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class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>不確定</span> </div> </a> <ul id="toc-不確定-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-機會率" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#機會率"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>機會率</span> </div> </a> <ul id="toc-機會率-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-概率公理" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#概率公理"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>概率公理</span> </div> </a> <ul id="toc-概率公理-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-重要概念" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#重要概念"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>重要概念</span> </div> </a> <button aria-controls="toc-重要概念-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>切換去重要概念細章節</span> </button> <ul id="toc-重要概念-sublist" class="vector-toc-list"> <li id="toc-交集同併集" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#交集同併集"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>交集同併集</span> </div> </a> <ul id="toc-交集同併集-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-對立同互斥" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#對立同互斥"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>對立同互斥</span> </div> </a> <ul id="toc-對立同互斥-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-條件概率" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#條件概率"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>條件概率</span> </div> </a> <ul id="toc-條件概率-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-獨立" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#獨立"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>獨立</span> </div> </a> <ul id="toc-獨立-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-概率分佈" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#概率分佈"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>概率分佈</span> </div> </a> <button aria-controls="toc-概率分佈-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>切換去概率分佈細章節</span> </button> <ul id="toc-概率分佈-sublist" class="vector-toc-list"> <li id="toc-伯努利分佈" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#伯努利分佈"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>伯努利分佈</span> </div> </a> <ul id="toc-伯努利分佈-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-常態分佈" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#常態分佈"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>常態分佈</span> </div> </a> <ul id="toc-常態分佈-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-隨機變數匯合" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#隨機變數匯合"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>隨機變數匯合</span> </div> </a> <button aria-controls="toc-隨機變數匯合-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>切換去隨機變數匯合細章節</span> </button> <ul id="toc-隨機變數匯合-sublist" class="vector-toc-list"> <li id="toc-大數定律" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#大數定律"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>大數定律</span> </div> </a> <ul id="toc-大數定律-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-中央極限定理" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#中央極限定理"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>中央極限定理</span> </div> </a> <ul id="toc-中央極限定理-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-概念詮釋" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#概念詮釋"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>概念詮釋</span> </div> </a> <button aria-controls="toc-概念詮釋-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>切換去概念詮釋細章節</span> </button> <ul id="toc-概念詮釋-sublist" class="vector-toc-list"> <li id="toc-古典定義" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#古典定義"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>古典定義</span> </div> </a> <ul id="toc-古典定義-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-物理思考" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#物理思考"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>物理思考</span> </div> </a> <ul id="toc-物理思考-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-統計應用" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#統計應用"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>統計應用</span> </div> </a> <ul id="toc-統計應用-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-睇埋" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#睇埋"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>睇埋</span> </div> </a> <ul id="toc-睇埋-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-文獻" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#文獻"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>文獻</span> </div> </a> <ul id="toc-文獻-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-註釋" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#註釋"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>註釋</span> </div> </a> <ul id="toc-註釋-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-詞彙" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#詞彙"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>詞彙</span> </div> </a> <ul id="toc-詞彙-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-引咗" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#引咗"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>引咗</span> </div> </a> <ul id="toc-引咗-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-拎" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#拎"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>拎</span> </div> </a> <ul id="toc-拎-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="目錄" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="開/收內容一覽" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">開/收內容一覽</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">概率論</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="去睇另一種語文嘅文章。有84種語言版本。" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-84" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">84種語言</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Waarskynlikheidsleer" title="Waarskynlikheidsleer – 南非荷蘭文" lang="af" hreflang="af" data-title="Waarskynlikheidsleer" data-language-autonym="Afrikaans" data-language-local-name="南非荷蘭文" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%D9%8A%D8%A9_%D8%A7%D9%84%D8%A7%D8%AD%D8%AA%D9%85%D8%A7%D9%84" title="نظرية الاحتمال – 阿拉伯文" lang="ar" hreflang="ar" data-title="نظرية الاحتمال" data-language-autonym="العربية" data-language-local-name="阿拉伯文" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Teor%C3%ADa_de_la_probabilid%C3%A1" title="Teoría de la probabilidá – 阿斯圖里亞文" lang="ast" hreflang="ast" data-title="Teoría de la probabilidá" data-language-autonym="Asturianu" data-language-local-name="阿斯圖里亞文" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Ehtimal_n%C9%99z%C9%99riyy%C9%99si" title="Ehtimal nəzəriyyəsi – 亞塞拜然文" lang="az" hreflang="az" data-title="Ehtimal nəzəriyyəsi" data-language-autonym="Azərbaycanca" data-language-local-name="亞塞拜然文" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%A7%D8%AD%D8%AA%DB%8C%D9%85%D8%A7%D9%84_%D9%86%D8%B8%D8%B1%DB%8C%D9%87%E2%80%8C%D8%B3%DB%8C" title="احتیمال نظریه‌سی – South Azerbaijani" lang="azb" hreflang="azb" data-title="احتیمال نظریه‌سی" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%98%D1%85%D1%82%D0%B8%D0%BC%D0%B0%D0%BB%D0%BB%D1%8B%D2%A1_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D2%BB%D1%8B" title="Ихтималлыҡ теорияһы – 巴什客爾文" lang="ba" hreflang="ba" data-title="Ихтималлыҡ теорияһы" data-language-autonym="Башҡортса" data-language-local-name="巴什客爾文" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A2%D1%8D%D0%BE%D1%80%D1%8B%D1%8F_%D1%96%D0%BC%D0%B0%D0%B2%D0%B5%D1%80%D0%BD%D0%B0%D1%81%D1%86%D0%B5%D0%B9" title="Тэорыя імавернасцей – 白俄羅斯文" lang="be" hreflang="be" data-title="Тэорыя імавернасцей" data-language-autonym="Беларуская" data-language-local-name="白俄羅斯文" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A2%D1%8D%D0%BE%D1%80%D1%8B%D1%8F_%D1%96%D0%BC%D0%B0%D0%B2%D0%B5%D1%80%D0%BD%D0%B0%D1%81%D1%8C%D1%86%D1%8F%D1%9E" title="Тэорыя імавернасьцяў – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Тэорыя імавернасьцяў" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%BD%D0%B0_%D0%B2%D0%B5%D1%80%D0%BE%D1%8F%D1%82%D0%BD%D0%BE%D1%81%D1%82%D0%B8%D1%82%D0%B5" title="Теория на вероятностите – 保加利亞文" lang="bg" hreflang="bg" data-title="Теория на вероятностите" data-language-autonym="Български" data-language-local-name="保加利亞文" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A6%AE%E0%A7%8D%E0%A6%AD%E0%A6%AC%E0%A6%A8%E0%A6%BE_%E0%A6%A4%E0%A6%A4%E0%A7%8D%E0%A6%A4%E0%A7%8D%E0%A6%AC" title="সম্ভবনা তত্ত্ব – 孟加拉文" lang="bn" hreflang="bn" data-title="সম্ভবনা তত্ত্ব" data-language-autonym="বাংলা" data-language-local-name="孟加拉文" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-btm mw-list-item"><a href="https://btm.wikipedia.org/wiki/Probabilitas" title="Probabilitas – Batak Mandailing" lang="btm" hreflang="btm" data-title="Probabilitas" data-language-autonym="Batak Mandailing" data-language-local-name="Batak Mandailing" class="interlanguage-link-target"><span>Batak Mandailing</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teoria_de_la_probabilitat" title="Teoria de la probabilitat – 加泰羅尼亞文" lang="ca" hreflang="ca" data-title="Teoria de la probabilitat" data-language-autonym="Català" data-language-local-name="加泰羅尼亞文" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AA%DB%8C%DB%86%D8%B1%DB%8C%DB%8C_%D8%A6%DB%95%DA%AF%DB%95%D8%B1" title="تیۆریی ئەگەر – 索拉尼庫爾德文" lang="ckb" hreflang="ckb" data-title="تیۆریی ئەگەر" data-language-autonym="کوردی" data-language-local-name="索拉尼庫爾德文" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Teorie_pravd%C4%9Bpodobnosti" title="Teorie pravděpodobnosti – 捷克文" lang="cs" hreflang="cs" data-title="Teorie pravděpodobnosti" data-language-autonym="Čeština" data-language-local-name="捷克文" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9F%D1%83%D0%BB%D0%B0%D1%8F%D1%81%D0%BB%C4%83%D1%85%D1%81%D0%B5%D0%BD_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D0%B9%C4%95" title="Пулаяслăхсен теорийĕ – 楚瓦什文" lang="cv" hreflang="cv" data-title="Пулаяслăхсен теорийĕ" data-language-autonym="Чӑвашла" data-language-local-name="楚瓦什文" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Damcaniaeth_tebygolrwydd" title="Damcaniaeth tebygolrwydd – 威爾斯文" lang="cy" hreflang="cy" data-title="Damcaniaeth tebygolrwydd" data-language-autonym="Cymraeg" data-language-local-name="威爾斯文" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Sandsynlighedsregning" title="Sandsynlighedsregning – 丹麥文" lang="da" hreflang="da" data-title="Sandsynlighedsregning" data-language-autonym="Dansk" data-language-local-name="丹麥文" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Wahrscheinlichkeitstheorie" title="Wahrscheinlichkeitstheorie – 德文" lang="de" hreflang="de" data-title="Wahrscheinlichkeitstheorie" data-language-autonym="Deutsch" data-language-local-name="德文" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%98%CE%B5%CF%89%CF%81%CE%AF%CE%B1_%CF%80%CE%B9%CE%B8%CE%B1%CE%BD%CE%BF%CF%84%CE%AE%CF%84%CF%89%CE%BD" title="Θεωρία πιθανοτήτων – 希臘文" lang="el" hreflang="el" data-title="Θεωρία πιθανοτήτων" data-language-autonym="Ελληνικά" data-language-local-name="希臘文" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Probability_theory" title="Probability theory – 英文" lang="en" hreflang="en" data-title="Probability theory" data-language-autonym="English" data-language-local-name="英文" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Probablo-teorio_(matematiko)" title="Probablo-teorio (matematiko) – 世界文" lang="eo" hreflang="eo" data-title="Probablo-teorio (matematiko)" data-language-autonym="Esperanto" data-language-local-name="世界文" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teor%C3%ADa_de_la_probabilidad" title="Teoría de la probabilidad – 西班牙文" lang="es" hreflang="es" data-title="Teoría de la probabilidad" data-language-autonym="Español" data-language-local-name="西班牙文" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/T%C3%B5en%C3%A4osusteooria" title="Tõenäosusteooria – 愛沙尼亞文" lang="et" hreflang="et" data-title="Tõenäosusteooria" data-language-autonym="Eesti" data-language-local-name="愛沙尼亞文" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Probabilitate_teoria" title="Probabilitate teoria – 巴斯克文" lang="eu" hreflang="eu" data-title="Probabilitate teoria" data-language-autonym="Euskara" data-language-local-name="巴斯克文" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%DB%8C%D9%87_%D8%A7%D8%AD%D8%AA%D9%85%D8%A7%D9%84%D8%A7%D8%AA" title="نظریه احتمالات – 波斯文" lang="fa" hreflang="fa" data-title="نظریه احتمالات" data-language-autonym="فارسی" data-language-local-name="波斯文" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Todenn%C3%A4k%C3%B6isyysteoria" title="Todennäköisyysteoria – 芬蘭文" lang="fi" hreflang="fi" data-title="Todennäköisyysteoria" data-language-autonym="Suomi" data-language-local-name="芬蘭文" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9orie_des_probabilit%C3%A9s" title="Théorie des probabilités – 法文" lang="fr" hreflang="fr" data-title="Théorie des probabilités" data-language-autonym="Français" data-language-local-name="法文" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Teoiric_na_d%C3%B3ch%C3%BAlachta" title="Teoiric na dóchúlachta – 愛爾蘭文" lang="ga" hreflang="ga" data-title="Teoiric na dóchúlachta" data-language-autonym="Gaeilge" data-language-local-name="愛爾蘭文" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Teor%C3%ADa_da_probabilidade" title="Teoría da probabilidade – 加里西亞文" lang="gl" hreflang="gl" data-title="Teoría da probabilidade" data-language-autonym="Galego" data-language-local-name="加里西亞文" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%94%D7%A1%D7%AA%D7%91%D7%A8%D7%95%D7%AA" title="תורת ההסתברות – 希伯來文" lang="he" hreflang="he" data-title="תורת ההסתברות" data-language-autonym="עברית" data-language-local-name="希伯來文" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%BE%E0%A4%AF%E0%A4%BF%E0%A4%95%E0%A4%A4%E0%A4%BE_%E0%A4%B8%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%BE%E0%A4%82%E0%A4%A4" title="प्रायिकता सिद्धांत – 北印度文" lang="hi" hreflang="hi" data-title="प्रायिकता सिद्धांत" data-language-autonym="हिन्दी" data-language-local-name="北印度文" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Teorija_vjerojatnosti" title="Teorija vjerojatnosti – 克羅埃西亞文" lang="hr" hreflang="hr" data-title="Teorija vjerojatnosti" data-language-autonym="Hrvatski" data-language-local-name="克羅埃西亞文" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Val%C3%B3sz%C3%ADn%C5%B1s%C3%A9gsz%C3%A1m%C3%ADt%C3%A1s" title="Valószínűségszámítás – 匈牙利文" lang="hu" hreflang="hu" data-title="Valószínűségszámítás" data-language-autonym="Magyar" data-language-local-name="匈牙利文" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%A1%D5%BE%D5%A1%D5%B6%D5%A1%D5%AF%D5%A1%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6%D5%B6%D5%A5%D6%80%D5%AB_%D5%BF%D5%A5%D5%BD%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Հավանականությունների տեսություն – 亞美尼亞文" lang="hy" hreflang="hy" data-title="Հավանականությունների տեսություն" data-language-autonym="Հայերեն" data-language-local-name="亞美尼亞文" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Teori_peluang" title="Teori peluang – 印尼文" lang="id" hreflang="id" data-title="Teori peluang" data-language-autonym="Bahasa Indonesia" data-language-local-name="印尼文" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/L%C3%ADkindafr%C3%A6%C3%B0i" title="Líkindafræði – 冰島文" lang="is" hreflang="is" data-title="Líkindafræði" data-language-autonym="Íslenska" data-language-local-name="冰島文" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teoria_della_probabilit%C3%A0" title="Teoria della probabilità – 義大利文" lang="it" hreflang="it" data-title="Teoria della probabilità" data-language-autonym="Italiano" data-language-local-name="義大利文" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%A2%BA%E7%8E%87%E8%AB%96" title="確率論 – 日文" lang="ja" hreflang="ja" data-title="確率論" data-language-autonym="日本語" data-language-local-name="日文" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/T%C3%A9ori_probabilitas" title="Téori probabilitas – 爪哇文" lang="jv" hreflang="jv" data-title="Téori probabilitas" data-language-autonym="Jawa" data-language-local-name="爪哇文" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%90%E1%83%9A%E1%83%91%E1%83%90%E1%83%97%E1%83%9D%E1%83%91%E1%83%98%E1%83%A1_%E1%83%97%E1%83%94%E1%83%9D%E1%83%A0%E1%83%98%E1%83%90" title="ალბათობის თეორია – 喬治亞文" lang="ka" hreflang="ka" data-title="ალბათობის თეორია" data-language-autonym="ქართული" data-language-local-name="喬治亞文" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/Itimall%C4%B1qlar_teoriyas%C4%B1" title="Itimallıqlar teoriyası – 卡拉卡爾帕克文" lang="kaa" hreflang="kaa" data-title="Itimallıqlar teoriyası" data-language-autonym="Qaraqalpaqsha" data-language-local-name="卡拉卡爾帕克文" class="interlanguage-link-target"><span>Qaraqalpaqsha</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%AB%D2%9B%D1%82%D0%B8%D0%BC%D0%B0%D0%BB%D0%B4%D1%8B%D2%9B_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D1%81%D1%8B" title="Ықтималдық теориясы – 哈薩克文" lang="kk" hreflang="kk" data-title="Ықтималдық теориясы" data-language-autonym="Қазақша" data-language-local-name="哈薩克文" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%99%95%EB%A5%A0%EB%A1%A0" title="확률론 – 韓文" lang="ko" hreflang="ko" data-title="확률론" data-language-autonym="한국어" data-language-local-name="韓文" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%AB%D0%BA%D1%82%D1%8B%D0%BC%D0%B0%D0%BB%D0%B4%D1%83%D1%83%D0%BB%D1%83%D0%BA_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D1%81%D1%8B" title="Ыктымалдуулук теориясы – 吉爾吉斯文" lang="ky" hreflang="ky" data-title="Ыктымалдуулук теориясы" data-language-autonym="Кыргызча" data-language-local-name="吉爾吉斯文" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Theoria_probabilitatum" title="Theoria probabilitatum – 拉丁文" lang="la" hreflang="la" data-title="Theoria probabilitatum" data-language-autonym="Latina" data-language-local-name="拉丁文" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Tikimybi%C5%B3_teorija" title="Tikimybių teorija – 立陶宛文" lang="lt" hreflang="lt" data-title="Tikimybių teorija" data-language-autonym="Lietuvių" data-language-local-name="立陶宛文" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Varb%C5%ABt%C4%ABbu_teorija" title="Varbūtību teorija – 拉脫維亞文" lang="lv" hreflang="lv" data-title="Varbūtību teorija" data-language-autonym="Latviešu" data-language-local-name="拉脫維亞文" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%98%D0%B0_%D0%BD%D0%B0_%D0%B2%D0%B5%D1%80%D0%BE%D1%98%D0%B0%D1%82%D0%BD%D0%BE%D1%81%D1%82%D0%B0" title="Теорија на веројатноста – 馬其頓文" lang="mk" hreflang="mk" data-title="Теорија на веројатноста" data-language-autonym="Македонски" data-language-local-name="馬其頓文" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%9C%D0%B0%D0%B3%D0%B0%D0%B4%D0%BB%D0%B0%D0%BB%D1%8B%D0%BD_%D0%BE%D0%BD%D0%BE%D0%BB" title="Магадлалын онол – 蒙古文" lang="mn" hreflang="mn" data-title="Магадлалын онол" data-language-autonym="Монгол" data-language-local-name="蒙古文" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Teori_kebarangkalian" title="Teori kebarangkalian – 馬來文" lang="ms" hreflang="ms" data-title="Teori kebarangkalian" data-language-autonym="Bahasa Melayu" data-language-local-name="馬來文" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Kansrekening" title="Kansrekening – 荷蘭文" lang="nl" hreflang="nl" data-title="Kansrekening" data-language-autonym="Nederlands" data-language-local-name="荷蘭文" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Sannsynsrekning" title="Sannsynsrekning – 耐諾斯克挪威文" lang="nn" hreflang="nn" data-title="Sannsynsrekning" data-language-autonym="Norsk nynorsk" data-language-local-name="耐諾斯克挪威文" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Sannsynlighetsteori" title="Sannsynlighetsteori – 巴克摩挪威文" lang="nb" hreflang="nb" data-title="Sannsynlighetsteori" data-language-autonym="Norsk bokmål" data-language-local-name="巴克摩挪威文" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Teoria_dei_probabilitats" title="Teoria dei probabilitats – 奧克西坦文" lang="oc" hreflang="oc" data-title="Teoria dei probabilitats" data-language-autonym="Occitan" data-language-local-name="奧克西坦文" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%B8%E0%AC%AE%E0%AD%8D%E0%AC%AD%E0%AC%BE%E0%AC%AC%E0%AD%8D%E0%AD%9F%E0%AC%A4%E0%AC%BE_%E0%AC%B8%E0%AC%BF%E0%AC%A6%E0%AD%8D%E0%AC%A7%E0%AC%BE%E0%AC%A8%E0%AD%8D%E0%AC%A4" title="ସମ୍ଭାବ୍ୟତା ସିଦ୍ଧାନ୍ତ – 歐利亞文" lang="or" hreflang="or" data-title="ସମ୍ଭାବ୍ୟତା ସିଦ୍ଧାନ୍ତ" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="歐利亞文" class="interlanguage-link-target"><span>ଓଡ଼ିଆ</span></a></li><li class="interlanguage-link interwiki-os mw-list-item"><a href="https://os.wikipedia.org/wiki/%D0%A3%C3%A6%D0%B2%C3%A6%D0%BD%D1%8B_%D1%82%D0%B5%D0%BE%D1%80%D0%B8" title="Уæвæны теори – 奧塞提文" lang="os" hreflang="os" data-title="Уæвæны теори" data-language-autonym="Ирон" data-language-local-name="奧塞提文" class="interlanguage-link-target"><span>Ирон</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B8%E0%A9%B0%E0%A8%AD%E0%A8%BE%E0%A8%B5%E0%A8%BF%E0%A8%95%E0%A8%A4%E0%A8%BE_%E0%A8%B8%E0%A8%BF%E0%A8%A7%E0%A8%BE%E0%A8%82%E0%A8%A4" title="ਸੰਭਾਵਿਕਤਾ ਸਿਧਾਂਤ – 旁遮普文" lang="pa" hreflang="pa" data-title="ਸੰਭਾਵਿਕਤਾ ਸਿਧਾਂਤ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="旁遮普文" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Teoria_prawdopodobie%C5%84stwa" title="Teoria prawdopodobieństwa – 波蘭文" lang="pl" hreflang="pl" data-title="Teoria prawdopodobieństwa" data-language-autonym="Polski" data-language-local-name="波蘭文" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teoria_das_probabilidades" title="Teoria das probabilidades – 葡萄牙文" lang="pt" hreflang="pt" data-title="Teoria das probabilidades" data-language-autonym="Português" data-language-local-name="葡萄牙文" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Teoria_probabilit%C4%83%C8%9Bilor" title="Teoria probabilităților – 羅馬尼亞文" lang="ro" hreflang="ro" data-title="Teoria probabilităților" data-language-autonym="Română" data-language-local-name="羅馬尼亞文" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%B2%D0%B5%D1%80%D0%BE%D1%8F%D1%82%D0%BD%D0%BE%D1%81%D1%82%D0%B5%D0%B9" title="Теория вероятностей – 俄文" lang="ru" hreflang="ru" data-title="Теория вероятностей" data-language-autonym="Русский" data-language-local-name="俄文" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Probability_theory" title="Probability theory – 蘇格蘭文" lang="sco" hreflang="sco" data-title="Probability theory" data-language-autonym="Scots" data-language-local-name="蘇格蘭文" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Teorija_vjerovatno%C4%87e" title="Teorija vjerovatnoće – 塞爾維亞克羅埃西亞文" lang="sh" hreflang="sh" data-title="Teorija vjerovatnoće" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="塞爾維亞克羅埃西亞文" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B7%83%E0%B6%B8%E0%B7%8A%E0%B6%B7%E0%B7%8F%E0%B7%80%E0%B7%92%E0%B6%AD%E0%B7%8F_%E0%B7%80%E0%B7%8F%E0%B6%AF%E0%B6%BA" title="සම්භාවිතා වාදය – 僧伽羅文" lang="si" hreflang="si" data-title="සම්භාවිතා වාදය" data-language-autonym="සිංහල" data-language-local-name="僧伽羅文" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Probability_theory" title="Probability theory – Simple English" lang="en-simple" hreflang="en-simple" data-title="Probability theory" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Te%C3%B3ria_pravdepodobnosti" title="Teória pravdepodobnosti – 斯洛伐克文" lang="sk" hreflang="sk" data-title="Teória pravdepodobnosti" data-language-autonym="Slovenčina" data-language-local-name="斯洛伐克文" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Verjetnostni_ra%C4%8Dun" title="Verjetnostni račun – 斯洛維尼亞文" lang="sl" hreflang="sl" data-title="Verjetnostni račun" data-language-autonym="Slovenščina" data-language-local-name="斯洛維尼亞文" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Teoria_e_probabilitetit" title="Teoria e probabilitetit – 阿爾巴尼亞文" lang="sq" hreflang="sq" data-title="Teoria e probabilitetit" data-language-autonym="Shqip" data-language-local-name="阿爾巴尼亞文" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%98%D0%B0_%D0%B2%D0%B5%D1%80%D0%BE%D0%B2%D0%B0%D1%82%D0%BD%D0%BE%D1%9B%D0%B5" title="Теорија вероватноће – 塞爾維亞文" lang="sr" hreflang="sr" data-title="Теорија вероватноће" data-language-autonym="Српски / srpski" data-language-local-name="塞爾維亞文" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/T%C3%A9ori_probabilitas" title="Téori probabilitas – 巽他文" lang="su" hreflang="su" data-title="Téori probabilitas" data-language-autonym="Sunda" data-language-local-name="巽他文" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Sannolikhetsteori" title="Sannolikhetsteori – 瑞典文" lang="sv" hreflang="sv" data-title="Sannolikhetsteori" data-language-autonym="Svenska" data-language-local-name="瑞典文" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A8%E0%AE%BF%E0%AE%95%E0%AE%B4%E0%AF%8D%E0%AE%A4%E0%AE%95%E0%AE%B5%E0%AF%81%E0%AE%95%E0%AF%8D_%E0%AE%95%E0%AF%8B%E0%AE%9F%E0%AF%8D%E0%AE%AA%E0%AE%BE%E0%AE%9F%E0%AF%81" title="நிகழ்தகவுக் கோட்பாடு – 坦米爾文" lang="ta" hreflang="ta" data-title="நிகழ்தகவுக் கோட்பாடு" data-language-autonym="தமிழ்" data-language-local-name="坦米爾文" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%9D%D0%B0%D0%B7%D0%B0%D1%80%D0%B8%D1%8F%D0%B8_%D1%8D%D2%B3%D1%82%D0%B8%D0%BC%D0%BE%D0%BB%D0%B8%D1%8F%D1%82" title="Назарияи эҳтимолият – 塔吉克文" lang="tg" hreflang="tg" data-title="Назарияи эҳтимолият" data-language-autonym="Тоҷикӣ" data-language-local-name="塔吉克文" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%A4%E0%B8%A9%E0%B8%8E%E0%B8%B5%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B8%99%E0%B9%88%E0%B8%B2%E0%B8%88%E0%B8%B0%E0%B9%80%E0%B8%9B%E0%B9%87%E0%B8%99" title="ทฤษฎีความน่าจะเป็น – 泰文" lang="th" hreflang="th" data-title="ทฤษฎีความน่าจะเป็น" data-language-autonym="ไทย" data-language-local-name="泰文" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tk mw-list-item"><a href="https://tk.wikipedia.org/wiki/%C3%84htimallyk_teori%C3%BDasy" title="Ähtimallyk teoriýasy – 土庫曼文" lang="tk" hreflang="tk" data-title="Ähtimallyk teoriýasy" data-language-autonym="Türkmençe" data-language-local-name="土庫曼文" class="interlanguage-link-target"><span>Türkmençe</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Teorya_ng_probabilidad" title="Teorya ng probabilidad – 塔加路族文" lang="tl" hreflang="tl" data-title="Teorya ng probabilidad" data-language-autonym="Tagalog" data-language-local-name="塔加路族文" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Olas%C4%B1l%C4%B1k_teorisi" title="Olasılık teorisi – 土耳其文" lang="tr" hreflang="tr" data-title="Olasılık teorisi" data-language-autonym="Türkçe" data-language-local-name="土耳其文" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D1%96%D1%8F_%D0%B9%D0%BC%D0%BE%D0%B2%D1%96%D1%80%D0%BD%D0%BE%D1%81%D1%82%D0%B5%D0%B9" title="Теорія ймовірностей – 烏克蘭文" lang="uk" hreflang="uk" data-title="Теорія ймовірностей" data-language-autonym="Українська" data-language-local-name="烏克蘭文" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%DB%8C%DB%82_%D8%A7%D8%AD%D8%AA%D9%85%D8%A7%D9%84" title="نظریۂ احتمال – 烏都文" lang="ur" hreflang="ur" data-title="نظریۂ احتمال" data-language-autonym="اردو" data-language-local-name="烏都文" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Ehtimollar_nazariyasi" title="Ehtimollar nazariyasi – 烏茲別克文" lang="uz" hreflang="uz" data-title="Ehtimollar nazariyasi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="烏茲別克文" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/L%C3%BD_thuy%E1%BA%BFt_x%C3%A1c_su%E1%BA%A5t" title="Lý thuyết xác suất – 越南文" lang="vi" hreflang="vi" data-title="Lý thuyết xác suất" data-language-autonym="Tiếng Việt" data-language-local-name="越南文" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%A6%82%E7%8E%87%E8%AE%BA" title="概率论 – 吳語" lang="wuu" hreflang="wuu" data-title="概率论" data-language-autonym="吴语" data-language-local-name="吳語" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%98%D7%A2%D7%90%D7%A8%D7%99%D7%A2_%D7%A4%D7%95%D7%9F_%D7%9E%D7%A9%D7%9E%D7%A2%D7%95%D7%AA%D7%93%D7%99%D7%A7%D7%99%D7%99%D7%98" title="טעאריע פון משמעותדיקייט – 意第緒文" lang="yi" hreflang="yi" data-title="טעאריע פון משמעותדיקייט" data-language-autonym="ייִדיש" data-language-local-name="意第緒文" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a 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src="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Oh_the_games_people_play_now..._%2838575282372%29.jpg/150px-Oh_the_games_people_play_now..._%2838575282372%29.jpg" decoding="async" width="150" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Oh_the_games_people_play_now..._%2838575282372%29.jpg/225px-Oh_the_games_people_play_now..._%2838575282372%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/00/Oh_the_games_people_play_now..._%2838575282372%29.jpg/300px-Oh_the_games_people_play_now..._%2838575282372%29.jpg 2x" data-file-width="2818" data-file-height="3373" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption"><a href="/wiki/%E8%BC%AA%E7%9B%A4" title="輪盤">輪盤</a>（上）、<a href="/wiki/%E6%8E%9F%E9%8A%80%E4%BB%94" title="掟銀仔">掟銀仔</a>（左下）同<a href="/wiki/%E6%93%B2%E9%AA%B0%E4%BB%94" class="mw-redirect" title="擲骰仔">擲骰仔</a>（右下）一般都俾人認為係<a href="/wiki/%E9%9A%A8%E6%A9%9F%E4%BA%8B%E4%BB%B6" class="mw-redirect" title="隨機事件">本質上隨機嘅事件</a>。</div></div></div></div> <p><b>概率論</b>（<a href="/wiki/Wikipedia:%E7%B2%B5%E6%96%87%E6%8B%BC%E9%9F%B3" title="Wikipedia:粵文拼音">粵拼</a>：<span class="plainlinks" style="display:inline-block"><a rel="nofollow" class="external text" href="http://yue.forvo.com/word/yue/概率論/#yue"><span>koi<span class="Jpingtone">3</span> leot<span class="Jpingtone">2</span> leon<span class="Jpingtone">6</span></span></a></span>；<a href="/wiki/%E8%8B%B1%E6%96%87" title="英文">英文</a>：<span lang="en"><b>probability theory</b></span>）係研究<a href="/wiki/%E6%A6%82%E7%8E%87" title="概率">概率</a>嘅一個<a href="/wiki/%E6%95%B8%E5%AD%B8%E7%90%86%E8%AB%96" title="數學理論">數學理論</a>。<a href="/wiki/%E6%A6%82%E7%8E%87" title="概率">概率</a>係指一件<a href="/wiki/%E9%9A%A8%E6%A9%9F%E4%BA%8B%E4%BB%B6" class="mw-redirect" title="隨機事件">事件</a><a href="/wiki/%E4%B8%8D%E7%A2%BA%E5%AE%9A" title="不確定">有幾可能</a>係真，1 代表件事實會發生，0 代表件事實唔會發生，而 0.5 就代表件事「有 50% 機會發生」；例如家陣<a href="/wiki/%E6%8E%9F%E9%8A%80%E4%BB%94" title="掟銀仔">掟一個銀仔</a>，假設個銀仔冇出千嘅話，應該會有 50% 機會出公、50% 機會出字，而呢件事嘅結果（公定字）原則上係冇可能預測嘅<sup id="cite_ref-feller1968_1-0" class="reference"><a href="#cite_note-feller1968-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>，反映咗<a href="/wiki/%E4%B8%8D%E7%A2%BA%E5%AE%9A" title="不確定">不確定</a><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>。 </p><p>喺概率論史上，「概率呢個數值<a href="/wiki/%E6%A6%82%E7%8E%87%E5%98%85%E8%A9%AE%E9%87%8B" title="概率嘅詮釋">要點樣理解</a>」係一個有相當爭議性嘅問題：喺最基本上，古典嘅<a href="/wiki/%E6%B1%BA%E5%AE%9A%E8%AB%96" title="決定論">決定論</a><sup id="cite_ref-det_3-0" class="reference"><a href="#cite_note-det-3"><span class="cite-bracket">[</span>英 1<span class="cite-bracket">]</span></a></sup>主張，如果一個觀察者喺而家呢一刻完美知道嗮<a href="/wiki/%E5%AE%87%E5%AE%99" title="宇宙">宇宙</a>嘅狀態（例：知道每粒<a href="/wiki/%E5%8E%9F%E5%AD%90" title="原子">原子</a>喺乜位置同以乜嘢<a href="/wiki/%E9%80%9F%E5%BA%A6" title="速度">速度</a>郁緊等等），佢將會有能力靠<a href="/wiki/%E7%89%A9%E7%90%86%E5%AE%9A%E5%BE%8B" title="物理定律">物理定律</a>－假設佢識嗮所需嘅<a href="/wiki/%E7%89%A9%E7%90%86" class="mw-redirect" title="物理">物理</a>知識－完美預測宇宙下一刻嘅狀態<sup id="cite_ref-laplacedemon_4-0" class="reference"><a href="#cite_note-laplacedemon-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>，所以概率淨係反映人類知識唔夠，而主張呢個觀點嘅人會話「人之所以預測唔到掟銀仔嘅結果，係因為人知唔嗮<a href="/wiki/%E9%A2%A8%E5%90%91" title="風向">風向</a>等嘅<a href="/wiki/%E8%B3%87%E8%A8%8A" title="資訊">資訊</a>」<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>。不過廿世紀<a href="/wiki/%E9%87%8F%E5%AD%90%E5%8A%9B%E5%AD%B8" title="量子力學">量子力學</a>研究指出，宇宙裏面有部份嘅事件似乎係本質上就冇可能完全準確噉預測嘅，人頂嗮櫳都淨係有得估呢啲事件發生嘅概率<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>。 </p><p>喺廿一世紀初，概率論經已成為咗一個重要嘅數學理論。例如<a href="/wiki/%E7%B5%B1%E8%A8%88%E5%AD%B8" title="統計學">統計學</a>就係建基於概率論嘅<sup id="cite_ref-feller1968_1-1" class="reference"><a href="#cite_note-feller1968-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>，而概率同相關概念喺<a href="/wiki/%E6%A9%9F%E6%A2%B0%E5%AD%B8%E7%BF%92" title="機械學習">機械學習</a>－教<a href="/wiki/%E4%BA%BA%E5%B7%A5%E6%99%BA%E8%83%BD" title="人工智能">人工智能</a><a href="/wiki/%E5%AD%B8%E7%BF%92" title="學習">學習</a>同處理不確定嘅技術<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>－同埋<a href="/wiki/%E9%81%8A%E6%88%B2%E8%A8%AD%E8%A8%88" title="遊戲設計">遊戲設計</a>－可能會涉及設計<a href="/wiki/%E6%A9%9F%E7%8E%87%E9%81%8A%E6%88%B2" title="機率遊戲">帶有隨機嘅遊戲</a><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>－上都有用。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="基礎"><span id=".E5.9F.BA.E7.A4.8E"></span>基礎</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%A6%82%E7%8E%87%E8%AB%96&action=edit&section=1" title="編輯小節: 基礎"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote">睇埋：<a href="/wiki/%E9%9A%A8%E6%A9%9F" title="隨機">隨機</a>同<a href="/wiki/%E8%B3%87%E8%A8%8A%E7%90%86%E8%AB%96" title="資訊理論">資訊理論</a></div> <div class="mw-heading mw-heading3"><h3 id="不確定"><span id=".E4.B8.8D.E7.A2.BA.E5.AE.9A"></span>不確定</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%A6%82%E7%8E%87%E8%AB%96&action=edit&section=2" title="編輯小節: 不確定"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Slot_machine.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Slot_machine.jpg/300px-Slot_machine.jpg" decoding="async" width="300" height="199" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Slot_machine.jpg/450px-Slot_machine.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Slot_machine.jpg/600px-Slot_machine.jpg 2x" data-file-width="3008" data-file-height="2000" /></a><figcaption>一部碌緊嘅<a href="/wiki/%E8%80%81%E8%99%8E%E6%A9%9F" title="老虎機">老虎機</a>；喺部機停之前，玩嘅人唔肯定部機將會出乜嘢圖案組合。</figcaption></figure> <div role="note" class="hatnote">内文：<a href="/wiki/%E4%B8%8D%E7%A2%BA%E5%AE%9A" title="不確定">不確定</a></div> <p>概率論嘅基礎係<a href="/wiki/%E4%B8%8D%E7%A2%BA%E5%AE%9A" title="不確定">不確定</a><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>英 2<span class="cite-bracket">]</span></a></sup>：响最基本上，不確定係指「一個個體<a href="/wiki/%E8%B3%87%E8%A8%8A" title="資訊">資訊</a>唔夠、唔能夠預測跟住會發生乜事」嘅情況。想像 </p> <ul><li>家陣做一場<a href="/wiki/%E9%9A%A8%E6%A9%9F%E5%AF%A6%E9%A9%97" class="mw-redirect" title="隨機實驗">實驗</a><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>註 1<span class="cite-bracket">]</span></a></sup>，場實驗會有 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> 個<a href="/wiki/%E5%8F%AF%E8%83%BD%E7%B5%90%E6%9E%9C" title="可能結果">可能結果</a>，而做嗰個人會將個實驗重複 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> 咁多次；</li> <li>包含場實驗嗰 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> 個可能結果嘅<a href="/wiki/%E9%9B%86" class="mw-redirect" title="集">集</a>，就係場實驗嘅<a href="/wiki/%E6%A8%A3%E6%9C%AC%E7%A9%BA%E9%96%93" title="樣本空間">樣本空間</a><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>英 3<span class="cite-bracket">]</span></a></sup>，</li> <li>個樣本空間嘅<a href="/wiki/%E5%86%AA%E9%9B%86" title="冪集">冪集</a>就包含咗嗰 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> 次實驗嘅可能結果組合（<a href="/wiki/%E9%9A%A8%E6%A9%9F%E4%BA%8B%E4%BB%B6" class="mw-redirect" title="隨機事件">事件</a>）。</li></ul> <p>舉個例說明，想像依家擲一粒六面嘅<a href="/wiki/%E9%AA%B0%E4%BB%94" title="骰仔">骰仔</a>（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0365f0b9f2721ed3ebb488a96d7348d978acf8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=6}"></span>），樣本空間 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=\{1,2,3,4,5,6\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>6</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=\{1,2,3,4,5,6\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb36a912c178876733db88c49e4823c2c60432cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.658ex; height:2.843ex;" alt="{\displaystyle s=\{1,2,3,4,5,6\}}"></span>；跟住有個人搵個唔<a href="/wiki/%E9%80%8F%E6%98%8E" title="透明">透明</a>嘅<a href="/wiki/%E9%AA%B0%E7%9B%85" class="mw-redirect" title="骰盅">骰盅</a>𢫏住咗三粒六面骰（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/918bb386a7ca6891255b62ef91ccc022883f3809" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=3}"></span>），然後係噉勁搖個骰盅，假設佢完全冇方法睇到粒骰仔（資訊唔夠），佢响攞開個骰盅之前就會經歷不確定，唔知嗰三粒骰仔擲到啲咩數字－嗰三粒骰可能係<a href="/wiki/%E5%AF%A6%E9%9A%9B%E6%95%B8%E5%80%BC" title="實際數值">擲到</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,3,5\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>5</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,3,5\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dff31162647fd4911bca0b34f555de99928d1cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.88ex; height:2.843ex;" alt="{\displaystyle \{1,3,5\}}"></span> （其中一件事件）又得，擲到 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{6,2,3\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>6</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{6,2,3\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05196f01fc86900cee1f4cf2ff164a3d4e33aad3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.88ex; height:2.843ex;" alt="{\displaystyle \{6,2,3\}}"></span> （另一件事件）... 又得<sup id="cite_ref-ross2010_13-0" class="reference"><a href="#cite_note-ross2010-13"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-pap1984_14-0" class="reference"><a href="#cite_note-pap1984-14"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>。 </p> <div class="mw-heading mw-heading3"><h3 id="機會率"><span id=".E6.A9.9F.E6.9C.83.E7.8E.87"></span>機會率</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%A6%82%E7%8E%87%E8%AB%96&action=edit&section=3" title="編輯小節: 機會率"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote">内文：<a href="/wiki/%E6%A9%9F%E6%9C%83%E7%8E%87" class="mw-redirect" title="機會率">機會率</a></div> <p><a href="/wiki/%E6%A6%82%E7%8E%87" title="概率">概率</a><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>英 4<span class="cite-bracket">]</span></a></sup>，又叫<a href="/wiki/%E6%A9%9F%E6%9C%83%E7%8E%87" class="mw-redirect" title="機會率">機會率</a>或者<a href="/wiki/%E6%88%96%E7%84%B6%E7%8E%87" class="mw-redirect" title="或然率">或然率</a>，可以噉樣想像：家陣有若干件可能嘅事件，而分析者同每一個可能嘅事件<a href="/wiki/%E5%87%BD%E6%95%B8" title="函數">都俾一個數值佢</a>；每件事件掕住嗰個數值就係嗰件事件嘅概率 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>，用日常用語講表示「件事件有幾大機會發生」－0 表示<b>實唔會發生</b>，1 表示<b>實會發生</b><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>註 2<span class="cite-bracket">]</span></a></sup>。响廿一世紀嘅概率論當中，啲人一般會用以下噉嘅<a href="/wiki/%E6%95%B8%E5%AD%B8%E7%AC%A6%E8%99%9F" title="數學符號">數學符號</a>嚟表示所講嘅嘢<sup id="cite_ref-bain1992_17-0" class="reference"><a href="#cite_note-bain1992-17"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>： </p> <ul><li>啲人一般會用「<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f264d19e21604793c6dc54f8044df454db82744" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.298ex; height:2.843ex;" alt="{\displaystyle P(A)}"></span>」或者「<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73acce4b7989bbfd543e36d2be5f6e433a9bbaf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.047ex; height:2.843ex;" alt="{\displaystyle \Pr(A)}"></span>」嚟代表「<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> 發生嘅概率」，</li> <li>而一場實驗嘅結果（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(i)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(i)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de02d30f303aef8641ffe02de0d367140bd4753d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.89ex; height:2.843ex;" alt="{\displaystyle f(i)}"></span>）可以用噉嘅方式表達<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>－ <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(i)={\begin{cases}p_{1}&{\text{if }}i=1,\\p_{2}&{\text{if }}i=2,\\p_{3}&{\text{if }}i=3,\\...\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>i</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>i</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(i)={\begin{cases}p_{1}&{\text{if }}i=1,\\p_{2}&{\text{if }}i=2,\\p_{3}&{\text{if }}i=3,\\...\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b6ac59b2813e6e7b4cc97a45acb6d08fe794eee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:22.493ex; height:11.176ex;" alt="{\displaystyle f(i)={\begin{cases}p_{1}&{\text{if }}i=1,\\p_{2}&{\text{if }}i=2,\\p_{3}&{\text{if }}i=3,\\...\end{cases}}}"></span></dd> <dd>－上述呢段符號意思係「<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fb857369fbc28dab4b5d76bb835e6f196b1d1cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.063ex; height:2.176ex;" alt="{\displaystyle i=1}"></span> 嘅概率係 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b58f22283ca46dd5da309cc34303b06a797783" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:2.009ex;" alt="{\displaystyle p_{1}}"></span>」、「<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd4e89f84cbdca577f73238abfb2fa18119c6eca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.063ex; height:2.176ex;" alt="{\displaystyle i=2}"></span> 嘅概率係 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43f1b08d7d69712872e051c2b33fdfa9f5d42319" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:2.009ex;" alt="{\displaystyle p_{2}}"></span>」... 呀噉；<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> 可以想像成（例如）「<a href="/wiki/%E6%93%B2%E9%AA%B0%E4%BB%94" class="mw-redirect" title="擲骰仔">擲骰仔</a>得到嘅數」<sup id="cite_ref-ross2010_13-1" class="reference"><a href="#cite_note-ross2010-13"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>。</dd></dl></li></ul> <figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/File:Fair_dice_probability_distribution.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Fair_dice_probability_distribution.svg/540px-Fair_dice_probability_distribution.svg.png" decoding="async" width="540" height="235" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Fair_dice_probability_distribution.svg/810px-Fair_dice_probability_distribution.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Fair_dice_probability_distribution.svg/1080px-Fair_dice_probability_distribution.svg.png 2x" data-file-width="576" data-file-height="251" /></a><figcaption></figcaption></figure> <p>用<a href="/wiki/%E5%9C%96%E5%83%8F" class="mw-redirect" title="圖像">圖像</a>嘅方式嚟諗嘅話，概率可以用好似上圖噉嘅<a href="/wiki/%E6%A6%82%E7%8E%87%E5%88%86%E4%BD%88" title="概率分佈">方式</a>嚟表達；想像 <a href="/wiki/X_%E8%BB%B8" class="mw-redirect" title="X 軸">X 軸</a>係 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>，表示「擲一粒六面骰仔得到嘅數」，而 <a href="/wiki/Y_%E8%BB%B8" class="mw-redirect" title="Y 軸">Y 軸</a>表示各件呢啲事件嘅相應 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> 值。假如粒骰仔冇出千，應該每個數出現嘅概率都係一樣嘅。 </p> <div class="mw-heading mw-heading3"><h3 id="概率公理"><span id=".E6.A6.82.E7.8E.87.E5.85.AC.E7.90.86"></span>概率公理</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%A6%82%E7%8E%87%E8%AB%96&action=edit&section=4" title="編輯小節: 概率公理"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote">内文：<a href="/wiki/%E6%A6%82%E7%8E%87%E5%85%AC%E7%90%86" title="概率公理">概率公理</a></div> <p>根據廿一世紀概率論當中嘅<a href="/wiki/%E6%A6%82%E7%8E%87%E5%85%AC%E7%90%86" title="概率公理">概率公理</a><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>英 5<span class="cite-bracket">]</span></a></sup>，以下呢三條原則係概率論嘅<a href="/wiki/%E5%85%AC%E7%90%86" title="公理">公理</a>，即係話概率論當咗呢幾句嘢係<a href="/wiki/%E4%B8%8D%E8%AD%89%E8%87%AA%E6%98%8E" title="不證自明">不證自明</a>嘅<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>： </p> <ul><li><b>第一公理</b>：一件事件嘅概率係一個非<a href="/wiki/%E8%B2%A0%E6%95%B8" title="負數">負數</a>嘅<a href="/wiki/%E5%AF%A6%E6%95%B8" title="實數">實數</a>（不過可以係 <a href="/wiki/0" title="0">0</a>）， <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(E)\in \mathbb {R} ,P(E)\geq 0\qquad \forall E\in F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> <mspace width="2em" /> <mi mathvariant="normal">∀</mi> <mi>E</mi> <mo>∈</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(E)\in \mathbb {R} ,P(E)\geq 0\qquad \forall E\in F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac12b631af7065f7f811d265b249a030f37484c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.769ex; height:2.843ex;" alt="{\displaystyle P(E)\in \mathbb {R} ,P(E)\geq 0\qquad \forall E\in F}"></span>，當中</dd></dl> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> 係指一件事件，而</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> 係指所有事件結合嘅集合。</li></ul></li> <li><b>第二公理</b>：「最少一件<a href="/wiki/%E5%9F%BA%E6%9C%AC%E4%BA%8B%E4%BB%B6" title="基本事件">基本事件</a><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>英 6<span class="cite-bracket">]</span></a></sup>（指淨係包含一個<a href="/wiki/%E5%8F%AF%E8%83%BD%E7%B5%90%E6%9E%9C" title="可能結果">可能結果</a>嘅<a href="/wiki/%E9%9A%A8%E6%A9%9F%E4%BA%8B%E4%BB%B6" class="mw-redirect" title="隨機事件">事件</a>）發生嘅概率」係 1， <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(\Omega )=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(\Omega )=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a66a50d3879f50fc0efb4fbd61cdc832b823640b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.494ex; height:2.843ex;" alt="{\displaystyle P(\Omega )=1}"></span></dd></dl> <ul><li>噉亦即係話，是但搵一件事件 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span>，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(E)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/687fe7f4688af755503fd00e7538f285e2a9954b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.33ex; height:2.843ex;" alt="{\displaystyle P(E)}"></span> 嘅數值頂嗮櫳都只會係 1，冇得大過 1。如果一場實驗當中有件事件 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac42446bcd2cbb76ec8fe2895635d328da22e26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{1}}"></span> 嘅概率係 1（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(E_{1})=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(E_{1})=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/427a264bab3a9bfc8f33b3e35100d45d80ddc137" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.585ex; height:2.843ex;" alt="{\displaystyle P(E_{1})=1}"></span>），場實驗就冇任何嘅不確定喺入面－<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac42446bcd2cbb76ec8fe2895635d328da22e26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{1}}"></span> 實會發生，而第啲事件嘅概率冚唪唥都會係 0，實唔會發生。</li></ul></li> <li><b>第三公理</b>：任何<a href="/wiki/%E5%8F%AF%E6%95%B8%E9%9B%86" title="可數集">可數</a>嘅事件<a href="/wiki/%E4%B8%8D%E4%BA%A4%E9%9B%86" title="不交集">不交集</a><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>註 3<span class="cite-bracket">]</span></a></sup>（詳情可以睇埋下面<a href="/wiki/%E4%BA%92%E6%96%A5%E4%BA%8B%E4%BB%B6" class="mw-redirect" title="互斥事件">互斥事件</a>）<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{1},E_{2},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{1},E_{2},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b591480f19455e3df6418a3b9d77f040d4247b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.33ex; height:2.509ex;" alt="{\displaystyle E_{1},E_{2},\ldots }"></span> 會滿足以下呢條式： <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\left(\bigcup _{i=1}^{\infty }E_{i}\right)=\sum _{i=1}^{\infty }P(E_{i}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <munderover> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\left(\bigcup _{i=1}^{\infty }E_{i}\right)=\sum _{i=1}^{\infty }P(E_{i}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47f22fe03df467b1d20785e5026bac39fabd9edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.941ex; height:7.509ex;" alt="{\displaystyle P\left(\bigcup _{i=1}^{\infty }E_{i}\right)=\sum _{i=1}^{\infty }P(E_{i}).}"></span></dd></dl> <ul><li>簡單講，即係話如果有若干件事件係冇可能同時發生嘅，噉「呢啲事件裏面是但一件發生嘅概率」等如呢啲事件各自嘅概率就噉<a href="/wiki/%E5%8A%A0%E7%B8%BD" title="加總">加埋</a>。</li></ul></li></ul> <div style="clear: both;"></div> <div class="mw-heading mw-heading2"><h2 id="重要概念"><span id=".E9.87.8D.E8.A6.81.E6.A6.82.E5.BF.B5"></span>重要概念</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%A6%82%E7%8E%87%E8%AB%96&action=edit&section=5" title="編輯小節: 重要概念"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote">睇埋：<a href="/wiki/%E6%A6%82%E7%8E%87%E5%8F%8A%E7%B5%B1%E8%A8%88%E5%AD%B8%E8%A9%9E%E5%BD%99%E8%A1%A8" class="mw-redirect" title="概率及統計學詞彙表">概率及統計學詞彙表</a></div> <div class="mw-heading mw-heading3"><h3 id="交集同併集"><span id=".E4.BA.A4.E9.9B.86.E5.90.8C.E4.BD.B5.E9.9B.86"></span>交集同併集</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%A6%82%E7%8E%87%E8%AB%96&action=edit&section=6" title="編輯小節: 交集同併集"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote">内文：<a href="/wiki/%E4%BA%A4%E9%9B%86" title="交集">交集</a>同<a href="/wiki/%E4%BD%B5%E9%9B%86" title="併集">併集</a></div> <div role="note" class="hatnote">睇埋：<a href="/wiki/%E6%BA%AB%E6%B0%8F%E5%9C%96" title="溫氏圖">溫氏圖</a></div> <p>响概率論當中，兩件事件之間嘅關係最基本上有兩種－<a href="/wiki/%E4%BA%A4%E9%9B%86" title="交集">交集</a><sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>英 7<span class="cite-bracket">]</span></a></sup>同<a href="/wiki/%E4%BD%B5%E9%9B%86" title="併集">併集</a><sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>英 8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-feller1968_1-2" class="reference"><a href="#cite_note-feller1968-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>： </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A{\mbox{ and }}B)=P(A\cap B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> and </mtext> </mstyle> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(A{\mbox{ and }}B)=P(A\cap B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96e9e55c9d5a52cf85fba0b550f6b3c3384db579" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.714ex; height:2.843ex;" alt="{\displaystyle P(A{\mbox{ and }}B)=P(A\cap B)}"></span> 代表「<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> <b><a href="/wiki/%E9%82%8F%E8%BC%AF%E8%88%87" title="邏輯與">同</a></b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> 都發生嘅概率」（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> 同 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> 嘅<a href="/wiki/%E4%BA%A4%E9%9B%86" title="交集">交集</a>），而</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A{\mbox{ or }}B)=P(A\cup B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> or </mtext> </mstyle> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(A{\mbox{ or }}B)=P(A\cup B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f94e87830efdd54552bac9a2a95406a62952700" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.04ex; height:2.843ex;" alt="{\displaystyle P(A{\mbox{ or }}B)=P(A\cup B)}"></span> 就代表「<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> <b><a href="/wiki/%E9%82%8F%E8%BC%AF%E6%88%96" title="邏輯或">或者</a></b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> 發生嘅概率」（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> 同 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> 嘅<a href="/wiki/%E4%BD%B5%E9%9B%86" title="併集">併集</a>）。</li></ul> <p>上述嘅概念可以用<a href="/wiki/%E6%BA%AB%E6%B0%8F%E5%9C%96" title="溫氏圖">溫氏圖</a><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>英 9<span class="cite-bracket">]</span></a></sup>嚟表達：一幅溫氏圖會有若干個波，每個波代表一件事件；兩個波之間嘅相交空間代表嗰兩個波所代表嗰兩件事件嘅<a href="/wiki/%E4%BA%A4%E9%9B%86" title="交集">交集</a><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup>。例如係以下呢幅溫氏圖噉，幅圖表示咗三件事件－<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> 同 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>，每件事件各有一個波代表（睇每個波掕住嗰個<a href="/wiki/%E7%BE%85%E9%A6%AC%E5%AD%97%E6%AF%8D" class="mw-redirect" title="羅馬字母">羅馬字母</a>），而三個波相交嘅空間（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cap B\cap C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo>∩</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cap B\cap C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24cedf001cf394b59a4cb1a939ab567910dfb415" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.439ex; height:2.176ex;" alt="{\displaystyle A\cap B\cap C}"></span>）就係表示「三件事都發生」嘅概率嘅<a href="/wiki/%E6%A6%82%E7%8E%87%E7%A9%BA%E9%96%93" title="概率空間">空間</a>。 </p> <div style="clear: both;"></div> <figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/File:Venn_diagram_ABC_BW_Explanation.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Venn_diagram_ABC_BW_Explanation.png/420px-Venn_diagram_ABC_BW_Explanation.png" decoding="async" width="420" height="397" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Venn_diagram_ABC_BW_Explanation.png/630px-Venn_diagram_ABC_BW_Explanation.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Venn_diagram_ABC_BW_Explanation.png/840px-Venn_diagram_ABC_BW_Explanation.png 2x" data-file-width="906" data-file-height="856" /></a><figcaption></figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="對立同互斥"><span id=".E5.B0.8D.E7.AB.8B.E5.90.8C.E4.BA.92.E6.96.A5"></span>對立同互斥</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%A6%82%E7%8E%87%E8%AB%96&action=edit&section=7" title="編輯小節: 對立同互斥"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote">内文：<a href="/wiki/%E5%B0%8D%E7%AB%8B%E4%BA%8B%E4%BB%B6" title="對立事件">對立事件</a>、<a href="/wiki/%E4%BA%92%E6%96%A5" title="互斥">互斥</a>同<a href="/wiki/%E9%9D%9E%E4%BA%92%E6%96%A5" class="mw-redirect" title="非互斥">非互斥</a></div> <p><a href="/wiki/%E5%B0%8D%E7%AB%8B%E4%BA%8B%E4%BB%B6" title="對立事件">對立事件</a>、<a href="/wiki/%E4%BA%92%E6%96%A5%E4%BA%8B%E4%BB%B6" class="mw-redirect" title="互斥事件">互斥事件</a>同<a href="/wiki/%E9%9D%9E%E4%BA%92%E6%96%A5%E4%BA%8B%E4%BB%B6" class="mw-redirect" title="非互斥事件">非互斥事件</a>係三個緊密相關嘅概念： </p> <ul><li><a href="/wiki/%E5%B0%8D%E7%AB%8B%E4%BA%8B%E4%BB%B6" title="對立事件">對立事件</a><sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>英 10<span class="cite-bracket">]</span></a></sup>：「<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> 嘅對立事件」（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98a12527148d6ed68adc91d9b419eb4b92d58ef6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.428ex; height:2.509ex;" alt="{\displaystyle A'}"></span> 或者 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d03504da9a53f91a23a003bbaede82cc3afafcab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.224ex; height:2.676ex;" alt="{\displaystyle A^{C}}"></span>）係指「<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> 冇發生」呢一件事件<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>。 <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A')=1-P(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(A')=1-P(A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb58ad2fe9b20ed9fe99d0ea42adceb27286ebd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.382ex; height:3.009ex;" alt="{\displaystyle P(A')=1-P(A)}"></span></dd></dl> <ul><li>例如上面嗰幅溫氏圖當中三個波以外嘅空間（三件事都冇發生）就係「<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{C}\cap B^{C}\cap C^{C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msup> <mo>∩</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msup> <mo>∩</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{C}\cap B^{C}\cap C^{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60ff8f24db7d5475222de71b57fb012a93f45384" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.914ex; height:2.676ex;" alt="{\displaystyle A^{C}\cap B^{C}\cap C^{C}}"></span>」。</li></ul></li> <li><a href="/wiki/%E4%BA%92%E6%96%A5%E4%BA%8B%E4%BB%B6" class="mw-redirect" title="互斥事件">互斥事件</a><sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>英 11<span class="cite-bracket">]</span></a></sup>：如果話「<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> 同 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> 係互斥事件」，即係話兩件事冇可能同時發生<sup id="cite_ref-miller2012_30-0" class="reference"><a href="#cite_note-miller2012-30"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup>－ <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A\cap B)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(A\cap B)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f22cd1c5c2bf0ec0d32ed076f95d8ba7db14b81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.905ex; height:2.843ex;" alt="{\displaystyle P(A\cap B)=0}"></span>，</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A\cup B)=P(A)+P(B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(A\cup B)=P(A)+P(B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03b5945043e1ac5caa5484b0a10a2d5f19d3ca40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.2ex; height:2.843ex;" alt="{\displaystyle P(A\cup B)=P(A)+P(B)}"></span>（<a href="/wiki/%E6%A6%82%E7%8E%87%E5%85%AC%E7%90%86" title="概率公理">概率公理</a>第三條）。</dd></dl> <ul><li>例如上面嗰幅溫氏圖當中嘅「<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{C}\cap B\cap C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msup> <mo>∩</mo> <mi>B</mi> <mo>∩</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{C}\cap B\cap C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e393c4549f23bbd75e2130b6166ad60c19051ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.92ex; height:2.676ex;" alt="{\displaystyle A^{C}\cap B\cap C}"></span>」同「<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cap B\cap C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo>∩</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cap B\cap C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24cedf001cf394b59a4cb1a939ab567910dfb415" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.439ex; height:2.176ex;" alt="{\displaystyle A\cap B\cap C}"></span>」就係互斥事件－兩個空間<a href="/wiki/%E4%B8%8D%E4%BA%A4%E9%9B%86" title="不交集">冇交集</a><sup id="cite_ref-miller2012_30-1" class="reference"><a href="#cite_note-miller2012-30"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup>。</li></ul></li> <li><a href="/wiki/%E9%9D%9E%E4%BA%92%E6%96%A5%E4%BA%8B%E4%BB%B6" class="mw-redirect" title="非互斥事件">非互斥事件</a><sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>英 12<span class="cite-bracket">]</span></a></sup>：如果話「<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> 同 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> 係非互斥事件」，即係話兩件事有可能同時發生－ <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A\cap B)>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(A\cap B)>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4ac8b7e69c7815fc32b5d82d61e98d18ee248f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.905ex; height:2.843ex;" alt="{\displaystyle P(A\cap B)>0}"></span>，</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A\cup B)=P(A)+P(B)-P(A\cap B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(A\cup B)=P(A)+P(B)-P(A\cap B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c8de5b03034abcef68b1d393382856497e295a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.685ex; height:2.843ex;" alt="{\displaystyle P(A\cup B)=P(A)+P(B)-P(A\cap B)}"></span><sup id="cite_ref-miller2012_30-2" class="reference"><a href="#cite_note-miller2012-30"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup>。</dd></dl> <ul><li>例如上面嗰幅溫氏圖當中嘅「<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>」同「<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>」就係非互斥事件－兩個空間之間有個大過 0 嘅<a href="/wiki/%E4%BA%A4%E9%9B%86" title="交集">交集</a>（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A\cap B)>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(A\cap B)>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4ac8b7e69c7815fc32b5d82d61e98d18ee248f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.905ex; height:2.843ex;" alt="{\displaystyle P(A\cap B)>0}"></span>）。</li></ul></li></ul> <div class="mw-heading mw-heading3"><h3 id="條件概率"><span id=".E6.A2.9D.E4.BB.B6.E6.A6.82.E7.8E.87"></span>條件概率</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%A6%82%E7%8E%87%E8%AB%96&action=edit&section=8" title="編輯小節: 條件概率"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote">内文：<a href="/wiki/%E6%A2%9D%E4%BB%B6%E6%A6%82%E7%8E%87" title="條件概率">條件概率</a></div> <div role="note" class="hatnote">睇埋：<a href="/wiki/%E8%B2%9D%E8%91%89%E6%96%AF%E5%AE%9A%E7%90%86" title="貝葉斯定理">貝葉斯定理</a></div> <p><a href="/wiki/%E6%A2%9D%E4%BB%B6%E6%A6%82%E7%8E%87" title="條件概率">條件概率</a><sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>英 13<span class="cite-bracket">]</span></a></sup>係指「如果一件事件發生咗，另外一件事件發生嘅概率」，例：「已知 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> 發生咗，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> 嘅條件概率」嘅數學符號係 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A\mid B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∣</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(A\mid B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f8f30f4da85b53901e0871eb41ed8827f511bb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.999ex; height:2.843ex;" alt="{\displaystyle P(A\mid B)}"></span></dd></dl> <p>呢個數值可以用以下呢條式計<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup>： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∣</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1804505c0de877b843b29d645f394a9615e726b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.578ex; height:6.509ex;" alt="{\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}}"></span></dd></dl> <p>舉個例說明，想像以下呢幅溫氏圖，每個空間掕住嗰個數表示嗰件事件發生嘅概率，例如「<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fa091eb428443c9c5c5fcf32a69d3665c89e00c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle B_{1}}"></span>、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/199944d59dcc18842dfd1deab6000a1d1dadcbae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle B_{2}}"></span> 同 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db17e43bc3f2416503df52a301a7d2779ab3a8ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle B_{3}}"></span> 都冇發生」嘅概率係 0.34... 等等，「已知 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/199944d59dcc18842dfd1deab6000a1d1dadcbae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle B_{2}}"></span> 發生咗，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> 嘅條件概率」（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A\mid B_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∣</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(A\mid B_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dae648106d63c1f6974d0bec889c36409d8c7bb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.053ex; height:2.843ex;" alt="{\displaystyle P(A\mid B_{2})}"></span>）係 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {0.12}{0.12+0.04}}=0.75}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>0.12</mn> <mrow> <mn>0.12</mn> <mo>+</mo> <mn>0.04</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0.75</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {0.12}{0.12+0.04}}=0.75}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/945e0742dd326210651f5e5254192a383d1ec843" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.178ex; height:5.343ex;" alt="{\displaystyle {\frac {0.12}{0.12+0.04}}=0.75}"></span>。</dd></dl> <div style="clear: both;"></div> <figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/File:Conditional_probability.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Conditional_probability.svg/480px-Conditional_probability.svg.png" decoding="async" width="480" height="340" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Conditional_probability.svg/720px-Conditional_probability.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Conditional_probability.svg/960px-Conditional_probability.svg.png 2x" data-file-width="815" data-file-height="578" /></a><figcaption></figcaption></figure> <p>如果 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> 同 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> 係<a href="/wiki/%E4%BA%92%E6%96%A5%E4%BA%8B%E4%BB%B6" class="mw-redirect" title="互斥事件">互斥事件</a>嘅話，噉 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}={\frac {0}{P(B)}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∣</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>0</mn> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}={\frac {0}{P(B)}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cc60d63854e130db48127ed83659a9959b4686a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.092ex; height:6.509ex;" alt="{\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}={\frac {0}{P(B)}}=0}"></span>。 </p> <div class="mw-heading mw-heading3"><h3 id="獨立"><span id=".E7.8D.A8.E7.AB.8B"></span>獨立</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%A6%82%E7%8E%87%E8%AB%96&action=edit&section=9" title="編輯小節: 獨立"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote">内文：<a href="/wiki/%E7%8D%A8%E7%AB%8B_(%E6%A6%82%E7%8E%87%E8%AB%96)" title="獨立 (概率論)">獨立 (概率論)</a></div> <div role="note" class="hatnote">睇埋：<a href="/wiki/%E8%B3%AD%E5%BE%92%E8%AC%AC%E8%AA%A4" title="賭徒謬誤">賭徒謬誤</a></div> <p><a href="/wiki/%E7%8D%A8%E7%AB%8B_(%E6%A6%82%E7%8E%87%E8%AB%96)" title="獨立 (概率論)">統計獨立</a><sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>英 14<span class="cite-bracket">]</span></a></sup>係指幾件事件之間唔會影響對方發生嘅概率。如果話「<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> 同 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> 呢兩件事件之間獨立」，噉以下呢條式成立<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup>： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A\mid B)=P(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∣</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(A\mid B)=P(A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a3ac3620d10642db6666d53f8ad7ebeb8ac72cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.395ex; height:2.843ex;" alt="{\displaystyle P(A\mid B)=P(A)}"></span>（「已知 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> 發生咗，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> 發生嘅概率依然係 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f264d19e21604793c6dc54f8044df454db82744" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.298ex; height:2.843ex;" alt="{\displaystyle P(A)}"></span> 咁多。」）</dd></dl> <p>噉根據條件概率嘅<a href="/wiki/%E5%AE%9A%E7%BE%A9" title="定義">定義</a>，亦即係話以下嘅<a href="/wiki/%E6%96%B9%E7%A8%8B%E5%BC%8F" title="方程式">方程式</a>會成立： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}=P(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∣</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}=P(A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/734e3d37043689d1308c54d87605484e3b6ccb0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.974ex; height:6.509ex;" alt="{\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}=P(A)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A\cap B)=P(A)P(B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(A\cap B)=P(A)P(B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/944fd278f8321f15de48f2454a3a3ccee3569bc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.359ex; height:2.843ex;" alt="{\displaystyle P(A\cap B)=P(A)P(B)}"></span></dd></dl> <p>就算將 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> 同 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> 喺以上呢啲式當中嘅位置互換，上述講嘅嘢不變。<a href="/wiki/%E6%A2%9D%E4%BB%B6%E7%8D%A8%E7%AB%8B" title="條件獨立">條件獨立</a><sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>英 15<span class="cite-bracket">]</span></a></sup>就係指一件事件唔會影響另外兩件事件之間嘅條件概率，即係話如果 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A|B,C)=P(A|C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(A|B,C)=P(A|C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a19e7c797d7dbd0c8756056e6498a8d3fdbab5c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.318ex; height:2.843ex;" alt="{\displaystyle P(A|B,C)=P(A|C)}"></span>，</dd></dl> <p>噉 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> 同 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> 就算係「喺 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> 之下條件獨立」<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>英 16<span class="cite-bracket">]</span></a></sup>，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\perp \!\!\!\perp B|C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>⊥</mo> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mo>⊥</mo> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\perp \!\!\!\perp B|C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d300825af2da80c8b1dda1e5635ac90f1515804c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.475ex; height:2.843ex;" alt="{\displaystyle (A\perp \!\!\!\perp B|C)}"></span> <sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup>。 </p> <div style="clear: both;"></div> <div class="mw-heading mw-heading2"><h2 id="概率分佈"><span id=".E6.A6.82.E7.8E.87.E5.88.86.E4.BD.88"></span>概率分佈</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%A6%82%E7%8E%87%E8%AB%96&action=edit&section=10" title="編輯小節: 概率分佈"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote">内文：<a href="/wiki/%E6%A6%82%E7%8E%87%E5%88%86%E4%BD%88" title="概率分佈">概率分佈</a></div> <p><a href="/wiki/%E6%A6%82%E7%8E%87%E5%88%86%E4%BD%88" title="概率分佈">概率分佈</a><sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>英 17<span class="cite-bracket">]</span></a></sup>係<a href="/wiki/%E7%B5%B1%E8%A8%88%E5%AD%B8" title="統計學">統計學</a>上成日用到嘅一樣嘢。一個概率分佈係一個表明某個<a href="/wiki/%E9%9A%A8%E6%A9%9F%E8%AE%8A%E6%95%B8" title="隨機變數">隨機變數</a>嘅每個可能數值出現嘅<a href="/wiki/%E6%A6%82%E7%8E%87" title="概率">概率</a>嘅<a href="/wiki/%E5%87%BD%E6%95%B8" title="函數">函數</a>， </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(X=x)=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>=</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(X=x)=f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de77f093e966fdeeb7fc3afbe68522eea9235c6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.228ex; height:2.843ex;" alt="{\displaystyle \Pr(X=x)=f(x)}"></span></dd></dl> <p>當中 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> 就係個概率分佈；個函數可以畫做一個表，<a href="/wiki/X_%E8%BB%B8" class="mw-redirect" title="X 軸">X 軸</a>代表個目標變數嘅數值，<a href="/wiki/Y_%E8%BB%B8" class="mw-redirect" title="Y 軸">Y 軸</a>代表嗰個目標變數嘅每個數值出現嘅概率<sup id="cite_ref-robert2008_41-0" class="reference"><a href="#cite_note-robert2008-41"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup>。 </p> <div class="mw-heading mw-heading3"><h3 id="伯努利分佈"><span id=".E4.BC.AF.E5.8A.AA.E5.88.A9.E5.88.86.E4.BD.88"></span>伯努利分佈</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%A6%82%E7%8E%87%E8%AB%96&action=edit&section=11" title="編輯小節: 伯努利分佈"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote">内文：<a href="/wiki/%E4%BC%AF%E5%8A%AA%E5%88%A9%E5%88%86%E4%BD%88" title="伯努利分佈">伯努利分佈</a></div> <p><a href="/wiki/%E4%BC%AF%E5%8A%AA%E5%88%A9%E5%88%86%E4%BD%88" title="伯努利分佈">伯努利分佈</a><sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>英 18<span class="cite-bracket">]</span></a></sup>就係一個可以用嚟模擬<a href="/wiki/%E6%8E%9F%E9%8A%80%E4%BB%94" title="掟銀仔">掟銀仔</a>嘅結果嘅概率分佈，一個伯努利分佈描述嘅變數 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> 得兩個可能數值，數值係 1 嘅機會率係 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>，數值係 0 嘅機會率係 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=(1-p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=(1-p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/402ed36ceea0cb5e046be9c353afa8d4944c46c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.15ex; height:2.843ex;" alt="{\displaystyle q=(1-p)}"></span>，即係 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(k;p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(k;p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f312a48a0ea3d81d8c1db50e80fc1846cfe7edc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.502ex; height:2.843ex;" alt="{\displaystyle f(k;p)}"></span> 係<sup id="cite_ref-bertsekas2002_43-0" class="reference"><a href="#cite_note-bertsekas2002-43"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup>： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(k;p)={\begin{cases}p&{\text{if }}k=1,\\q=1-p&{\text{if }}k=0.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>p</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>q</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>k</mi> <mo>=</mo> <mn>0.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(k;p)={\begin{cases}p&{\text{if }}k=1,\\q=1-p&{\text{if }}k=0.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fff3412509e73816dc2b28405b93c34f89ee487" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.817ex; height:6.176ex;" alt="{\displaystyle f(k;p)={\begin{cases}p&{\text{if }}k=1,\\q=1-p&{\text{if }}k=0.\end{cases}}}"></span></dd></dl> <p>當中 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> 可以等如 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>（兩個都係 0.5）。假想而家用一個伯努利分佈嚟模擬一次<a href="/wiki/%E6%8E%9F%E9%8A%80%E4%BB%94" title="掟銀仔">掟銀仔</a>實驗嘅結果，個銀仔俾人做咗手腳，所以出公嘅概率（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>）同出字嘅概率（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>）唔相同，畫做圖（X 軸表示掟銀仔結果嘅數值、Y 軸表示每個數值出現嘅概率）嘅話就會好似以下呢幅圖噉樣： </p> <div style="clear: both;"></div> <figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/File:Bernuilli_distribution_PDF.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Bernuilli_distribution_PDF.svg/450px-Bernuilli_distribution_PDF.svg.png" decoding="async" width="450" height="338" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Bernuilli_distribution_PDF.svg/675px-Bernuilli_distribution_PDF.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Bernuilli_distribution_PDF.svg/900px-Bernuilli_distribution_PDF.svg.png 2x" data-file-width="325" data-file-height="244" /></a><figcaption></figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="常態分佈"><span id=".E5.B8.B8.E6.85.8B.E5.88.86.E4.BD.88"></span>常態分佈</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%A6%82%E7%8E%87%E8%AB%96&action=edit&section=12" title="編輯小節: 常態分佈"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote">内文：<a href="/wiki/%E5%B8%B8%E6%85%8B%E5%88%86%E4%BD%88" title="常態分佈">常態分佈</a></div> <p>响現實世界嘅<a href="/wiki/%E7%A7%91%E7%A0%94" class="mw-redirect" title="科研">科研</a>入面，啲變數好少可會「一係公一係字」咁二元，但個原理一樣：<a href="/wiki/%E5%B8%B8%E6%85%8B%E5%88%86%E4%BD%88" title="常態分佈">常態分佈</a><sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>英 19<span class="cite-bracket">]</span></a></sup>就係科學入面最常用嘅概率分佈之一；常態分佈模擬嘅係一個<a href="/wiki/%E9%80%A3%E7%BA%8C%E8%AE%8A%E6%95%B8" class="mw-redirect" title="連續變數">連續變數</a>（即係個目標變數嘅數值喺<a href="/wiki/%E5%B0%8F%E6%95%B8%E9%BB%9E" class="mw-redirect" title="小數點">小數點</a>裏面有幾多個位都得），而喺一個常態分佈當中，個變數嘅<a href="/wiki/%E5%B9%B3%E5%9D%87%E5%80%BC" title="平均值">平均值</a>會係出現得最密嘅數值，低過平均嘅數值同高過平均嘅數值出現嘅機會率一樣，而離平均值愈遠嘅數值，抽到出嚟嘅機會率就愈低，如果按住個樣本畫一個概率分佈圖，一個常態分佈會俾出一條好似<a href="/wiki/%E9%90%98%E5%BD%A2%E6%9B%B2%E7%B7%9A" class="mw-redirect" title="鐘形曲線">鐘噉嘅形狀嘅線</a>。常態分佈嘅<a href="/wiki/%E6%A6%82%E7%8E%87%E5%AF%86%E5%BA%A6%E5%87%BD%E6%95%B8" title="概率密度函數">概率密度函數</a>係（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"></span> 係個分佈嘅<a href="/wiki/%E6%A8%99%E6%BA%96%E5%B7%AE" title="標準差">標準差</a>）<sup id="cite_ref-bryc_45-0" class="reference"><a href="#cite_note-bryc-45"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup>： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mi>μ</mi> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00cb9b2c9b866378626bcfa45c86a6de2f2b2e40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:24.446ex; height:6.676ex;" alt="{\displaystyle f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}}"></span></dd></dl> <div style="clear: both;"></div> <figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/File:Standard_deviation_diagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Standard_deviation_diagram.svg/570px-Standard_deviation_diagram.svg.png" decoding="async" width="570" height="285" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Standard_deviation_diagram.svg/855px-Standard_deviation_diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Standard_deviation_diagram.svg/1140px-Standard_deviation_diagram.svg.png 2x" data-file-width="400" data-file-height="200" /></a><figcaption></figcaption></figure> <p>譬如話以下呢個情況噉：想像有<a href="/wiki/%E7%94%9F%E7%89%A9%E5%AD%B8%E5%AE%B6" title="生物學家">生物學家</a>想研究成年<a href="/wiki/%E4%B8%AD%E8%8F%AF%E7%99%BD%E6%B5%B7%E8%B1%9A" title="中華白海豚">中華白海豚</a>嘅身長，但佢冇可能捉嗮世界上咁多隻白海豚遂隻遂隻嚟度佢哋幾長，於是乎佢就抽個樣本出嚟，用個樣本嚟估計全世界嘅白海豚嘅身長；呢個樣本入面有 20 隻白海豚，佢哋嘅平均身長係 2.2 米，唔係隻隻都啱啱好 2.2 米長－有隻係 1.8 米長，有隻係 2.6 米長呀噉－但一隻白海豚身長高過平均身長嘅機會率大致上等如佢身長低過平均嘅機會率，而且離 2.2 米愈遠嘅數值出現嘅機會率愈低。如果畫幅概率分佈圖，「隻白海豚嘅身長」做 X 軸，而「每個身長數值出現嘅機會率」做 Y 軸，幅圖會出一條近似鐘形嘅線。實證嘅科學研究經已表明咗，<a href="/wiki/%E6%99%BA%E5%95%86" title="智商">智商</a>（IQ）等好多重要嘅變數嘅分佈都可以用常態分佈嚟模擬<sup id="cite_ref-bryc_45-1" class="reference"><a href="#cite_note-bryc-45"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup>。 </p> <div class="mw-heading mw-heading2"><h2 id="隨機變數匯合"><span id=".E9.9A.A8.E6.A9.9F.E8.AE.8A.E6.95.B8.E5.8C.AF.E5.90.88"></span>隨機變數匯合</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%A6%82%E7%8E%87%E8%AB%96&action=edit&section=13" title="編輯小節: 隨機變數匯合"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote">内文：<a href="/wiki/%E9%9A%A8%E6%A9%9F%E8%AE%8A%E6%95%B8%E5%8C%AF%E5%90%88" title="隨機變數匯合">隨機變數匯合</a></div> <p><a href="/wiki/%E9%9A%A8%E6%A9%9F%E8%AE%8A%E6%95%B8%E5%8C%AF%E5%90%88" title="隨機變數匯合">隨機變數匯合</a><sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>英 20<span class="cite-bracket">]</span></a></sup>係指隨機變數有嘅<a href="/wiki/%E6%A5%B5%E9%99%90_(%E5%87%BD%E6%95%B8)" title="極限 (函數)">極限</a><sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>英 21<span class="cite-bracket">]</span></a></sup>。簡單講，如果話某一個隨機變數 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> 有一個極限，即係指（例如）隨住某個數值 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> 變得愈嚟愈大，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> 嘅數值會慢慢愈嚟愈近（匯合）某個數值（設呢個數值做 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span>，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> 係個函數嘅極限）<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup>－ </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }x=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>x</mi> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }x=c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc21d1b368bfa656bbb0e36569d6b08720046929" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.095ex; height:3.676ex;" alt="{\displaystyle \lim _{n\to \infty }x=c}"></span></dd></dl> <p>喺概率論上，隨機變數匯合相關嘅現象有以下呢啲： </p> <div class="mw-heading mw-heading3"><h3 id="大數定律"><span id=".E5.A4.A7.E6.95.B8.E5.AE.9A.E5.BE.8B"></span>大數定律</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%A6%82%E7%8E%87%E8%AB%96&action=edit&section=14" title="編輯小節: 大數定律"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote">内文：<a href="/wiki/%E5%A4%A7%E6%95%B8%E5%AE%9A%E5%BE%8B" title="大數定律">大數定律</a></div> <p><a href="/wiki/%E5%A4%A7%E6%95%B8%E5%AE%9A%E5%BE%8B" title="大數定律">大數定律</a><sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>英 22<span class="cite-bracket">]</span></a></sup>係概率論上一條俾人覺得好合乎<a href="/wiki/%E7%9B%B4%E8%A6%BA" title="直覺">直覺</a>嘅定律<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup>：想像家陣掟一個冇做手腳（出公出字概率一樣）嘅<a href="/wiki/%E9%8A%80%E4%BB%94" title="銀仔">銀仔</a>，掟嘅次數係 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>；一般直覺認為，如果 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> 嘅數值極大（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d55d9b32f6fa8fab6a84ea444a6b5a24bb45e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\displaystyle n\to \infty }"></span>），噉嗰 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> 次掟銀仔嘅結果應該會有一半係公一半係字；而且 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> 嘅數值愈大，<b>出公嘅次數</b>同<b>出字嘅次數</b>之間嘅<a href="/wiki/%E6%AF%94%E4%BE%8B" title="比例">比例</a>應該會愈嚟愈接近 1。又想像家陣擲一粒（冇做手腳嘅）六面骰仔擲 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> 咁多次，噉嗰 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> 次擲骰仔嘅結果嘅<a href="/wiki/%E5%B9%B3%E5%9D%87%E5%80%BC" title="平均值">平均值</a>（<a href="/wiki/%E6%A8%A3%E6%9C%AC%E5%B9%B3%E5%9D%87%E5%80%BC" class="mw-redirect" title="樣本平均值">樣本平均值</a><sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>英 23<span class="cite-bracket">]</span></a></sup>）正路會隨住 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> 變得愈嚟愈大，而接近 3.5（理論上嘅平均值）－ </p> <figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/File:Largenumbers.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Largenumbers.png/480px-Largenumbers.png" decoding="async" width="480" height="384" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/0/02/Largenumbers.png 1.5x" data-file-width="600" data-file-height="480" /></a><figcaption></figcaption></figure> <p>喺比較嚴格嘅<a href="/wiki/%E5%AE%9A%E7%BE%A9" title="定義">定義</a>上，大數定律講嘅嘢如下：依家有一連串<a href="/wiki/%E7%8D%A8%E7%AB%8B%E5%90%8C%E5%88%86%E4%BD%88" title="獨立同分佈">獨立同分佈</a>（iid）嘅隨機變數 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33c25229c6989c235f9cbb7908331f6d01d0abfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.013ex; height:2.509ex;" alt="{\displaystyle X_{k}}"></span>（意思即係指呢啲變數之間<a href="/wiki/%E7%8D%A8%E7%AB%8B_(%E6%A6%82%E7%8E%87%E8%AB%96)" title="獨立 (概率論)">獨立</a>，而且<a href="/wiki/%E6%A6%82%E7%8E%87%E5%88%86%E4%BD%88" title="概率分佈">概率分佈</a>一樣）；只要 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |X_{k}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |X_{k}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d00970763c764e0beb2d9ae1e934f0d07de648af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.307ex; height:2.843ex;" alt="{\displaystyle |X_{k}|}"></span>（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33c25229c6989c235f9cbb7908331f6d01d0abfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.013ex; height:2.509ex;" alt="{\displaystyle X_{k}}"></span> 嘅<a href="/wiki/%E6%9C%9F%E6%9C%9B%E5%80%BC" title="期望值">期望值</a>）唔係<a href="/wiki/%E7%84%A1%E9%99%90%E5%A4%A7" class="mw-redirect" title="無限大">無限大</a>，噉 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33c25229c6989c235f9cbb7908331f6d01d0abfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.013ex; height:2.509ex;" alt="{\displaystyle X_{k}}"></span> 實際觀察到嘅樣本平均值（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {X}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X}}_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55a63d07fe7c766a92154413faf8456d31c29fca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.35ex; height:3.343ex;" alt="{\displaystyle {\overline {X}}_{n}}"></span>） </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {X}}_{n}={\frac {1}{n}}{\sum _{k=1}^{n}X_{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X}}_{n}={\frac {1}{n}}{\sum _{k=1}^{n}X_{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfdf015b3311f4d3fab8f2a67d0e53a8dc50a3f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.434ex; height:6.843ex;" alt="{\displaystyle {\overline {X}}_{n}={\frac {1}{n}}{\sum _{k=1}^{n}X_{k}}}"></span></dd></dl> <p>會隨住 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> 變大而接近 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |X_{k}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |X_{k}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d00970763c764e0beb2d9ae1e934f0d07de648af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.307ex; height:2.843ex;" alt="{\displaystyle |X_{k}|}"></span>。大數定律嘅諗頭源於直覺，但喺實際嘅觀察上經已受到證實，進階嘅分析仲會分<a href="/wiki/%E5%BC%B1%E5%A4%A7%E6%95%B8%E5%AE%9A%E5%BE%8B" class="mw-redirect" title="弱大數定律">弱大數定律</a>－「趨近樣本平均值有咁上下概率會發生」同<a href="/wiki/%E5%BC%B7%E5%A4%A7%E6%95%B8%E5%AE%9A%E5%BE%8B" class="mw-redirect" title="強大數定律">強大數定律</a>－「趨近樣本平均值係一定會發生咁滯」<sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup>。 </p> <div class="mw-heading mw-heading3"><h3 id="中央極限定理"><span id=".E4.B8.AD.E5.A4.AE.E6.A5.B5.E9.99.90.E5.AE.9A.E7.90.86"></span>中央極限定理</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%A6%82%E7%8E%87%E8%AB%96&action=edit&section=15" title="編輯小節: 中央極限定理"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote">内文：<a href="/wiki/%E4%B8%AD%E5%A4%AE%E6%A5%B5%E9%99%90%E5%AE%9A%E7%90%86" title="中央極限定理">中央極限定理</a></div> <p><a href="/wiki/%E4%B8%AD%E5%A4%AE%E6%A5%B5%E9%99%90%E5%AE%9A%E7%90%86" title="中央極限定理">中央極限定理</a><sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>英 24<span class="cite-bracket">]</span></a></sup>廣泛噉俾人認為係現代<a href="/wiki/%E6%95%B8%E5%AD%B8" title="數學">數學</a>上嘅一個重要結果，可以解釋<a href="/wiki/%E5%B8%B8%E6%85%8B%E5%88%86%E4%BD%88" title="常態分佈">常態分佈</a>（睇返上面）點解喺<a href="/wiki/%E5%A4%A7%E8%87%AA%E7%84%B6" class="mw-redirect" title="大自然">大自然</a>入面周圍都有（智商同<a href="/wiki/%E8%BA%AB%E9%AB%98" title="身高">身高</a>等好多表示個體差異嘅變數都呈常態分佈）。根據中央極限定理，假想有若干（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>）個<a href="/wiki/%E7%8D%A8%E7%AB%8B%E5%90%8C%E5%88%86%E4%BD%88" title="獨立同分佈">獨立同分佈</a>、<a href="/wiki/%E8%AE%8A%E7%95%B0%E6%95%B8" class="mw-redirect" title="變異數">變異數</a>有限嘅隨機變數，如果 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> 嘅數值極大（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d55d9b32f6fa8fab6a84ea444a6b5a24bb45e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\displaystyle n\to \infty }"></span>），噉呢啲隨機變數嘅<a href="/wiki/%E5%B9%B3%E5%9D%87%E5%80%BC" title="平均值">平均值</a>形成嘅分佈會接近一個常態分佈，無論啲隨機變數本身嘅分佈係點嘅樣都一樣。用例子說明嘅話，即係話想像 </p> <ul><li>家陣擲 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> 咁多粒骰仔，</li> <li>啲骰仔冇出千（<b>一粒骰仔擲到嘅數</b>個概率唔成常態分佈），</li> <li>根據中央極限定理，如果 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> 嘅數值極大，噉最後嗰 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> 粒骰仔<a href="/wiki/%E6%A8%A3%E6%9C%AC%E5%B9%B3%E5%9D%87%E5%80%BC" class="mw-redirect" title="樣本平均值">擲到嘅數嘅平均值</a>會成一個常態分佈<sup id="cite_ref-baranv2007_54-0" class="reference"><a href="#cite_note-baranv2007-54"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-dinov2008_55-0" class="reference"><a href="#cite_note-dinov2008-55"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup>。</li></ul> <p>用圖嚟表達嘅話： </p> <div style="clear: both;"></div> <ul class="gallery mw-gallery-packed center"> <li class="gallerybox" style="width: 842.66666666667px"> <div class="thumb" style="width: 840.66666666667px;"><span typeof="mw:File"><a href="/wiki/File:CLTBinomConvergence.svg" class="mw-file-description" title="唔同 '"`UNIQ--postMath-0000008A-QINU`"' 同 '"`UNIQ--postMath-0000008B-QINU`"' [註 4]值嘅二項分佈出嘅樣本平均值嘅分佈；綠線：理論上嘅常態分佈（用嚟做對照）；紅線：嗰個 '"`UNIQ--postMath-0000008C-QINU`"' 同 '"`UNIQ--postMath-0000008D-QINU`"' 值組合出嘅樣本平均值分佈。"><img alt="唔同 '"`UNIQ--postMath-0000008A-QINU`"' 同 '"`UNIQ--postMath-0000008B-QINU`"' [註 4]值嘅二項分佈出嘅樣本平均值嘅分佈；綠線：理論上嘅常態分佈（用嚟做對照）；紅線：嗰個 '"`UNIQ--postMath-0000008C-QINU`"' 同 '"`UNIQ--postMath-0000008D-QINU`"' 值組合出嘅樣本平均值分佈。" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/CLTBinomConvergence.svg/1261px-CLTBinomConvergence.svg.png" decoding="async" width="841" height="540" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/CLTBinomConvergence.svg/1892px-CLTBinomConvergence.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/CLTBinomConvergence.svg/2522px-CLTBinomConvergence.svg.png 2x" data-file-width="1012" data-file-height="650" /></a></span></div> <div class="gallerytext">唔同 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> 同 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> <sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>註 4<span class="cite-bracket">]</span></a></sup>值嘅<a href="/wiki/%E4%BA%8C%E9%A0%85%E5%88%86%E4%BD%88" title="二項分佈">二項分佈</a>出嘅<a href="/wiki/%E6%A8%A3%E6%9C%AC%E5%B9%B3%E5%9D%87%E5%80%BC" class="mw-redirect" title="樣本平均值">樣本平均值</a>嘅分佈；<br /><span style="color:green;">綠線</span>：理論上嘅常態分佈（用嚟做對照）；<br /><span style="color:green;">紅線</span>：嗰個 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> 同 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> 值組合出嘅樣本平均值分佈。</div> </li> </ul> <p>簡單噉講，設 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},\dots \,X_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mspace width="thinmathspace" /> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},\dots \,X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4057e8904d3af73af28542754bd80dd9823ba6cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.665ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},\dots \,X_{n}}"></span> 做 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> 個<a href="/wiki/%E7%8D%A8%E7%AB%8B%E5%90%8C%E5%88%86%E4%BD%88" title="獨立同分佈">獨立同分佈</a>嘅隨機變數，<a href="/wiki/%E5%B9%B3%E5%9D%87%E5%80%BC" title="平均值">平均值</a>係 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> 而<a href="/wiki/%E8%AE%8A%E7%95%B0%E6%95%B8" class="mw-redirect" title="變異數">變異數</a>係 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="{\displaystyle \sigma ^{2}}"></span>，設 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {X}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {X}}_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ab9d44cdd15adc4e2e00c9b282832eefe22d2dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.198ex; height:2.843ex;" alt="{\displaystyle {\bar {X}}_{n}}"></span> 做呢柞隨機變數嘅<a href="/wiki/%E5%B9%B3%E5%9D%87%E5%80%BC" title="平均值">平均值</a>，噉 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=\lim _{n\to \infty }{\sqrt {n}}{\left({\frac {{\bar {X}}_{n}-\mu }{\sigma }}\right)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z=\lim _{n\to \infty }{\sqrt {n}}{\left({\frac {{\bar {X}}_{n}-\mu }{\sigma }}\right)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5125e0eb5893199f1c18b4a9b98228b14f98edff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.467ex; height:6.343ex;" alt="{\displaystyle Z=\lim _{n\to \infty }{\sqrt {n}}{\left({\frac {{\bar {X}}_{n}-\mu }{\sigma }}\right)}}"></span></dd></dl> <p>會係一個常態分佈。中央極限定理講嘅嘢表示，大自然會充滿咗常態分佈－每一件<a href="/wiki/%E6%95%B8%E6%93%9A" title="數據">數據</a>（例：<a href="/wiki/%E8%A8%8A%E8%99%9F" title="訊號">訊號</a>當中嘅<a href="/wiki/%E9%9B%9C%E8%A8%8A" class="mw-redirect" title="雜訊">雜訊</a>）產生嘅過程可以想像成一個<a href="/wiki/%E9%9A%A8%E6%A9%9F%E9%81%8E%E7%A8%8B" title="隨機過程">隨機過程</a>，個結果（件雜訊搞到個訊號高咗定低咗，同埋搞到個訊號變咗幾多）會係一個隨機變數，而無論件數據嘅產生過程係乜嘢分佈都好，產生大量嘅數據（睇咗大量嘅訊號）之後，件數據（啲雜訊）嘅數值嘅平均值都會呈現常態分佈<sup id="cite_ref-baranv2007_54-1" class="reference"><a href="#cite_note-baranv2007-54"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-dinov2008_55-1" class="reference"><a href="#cite_note-dinov2008-55"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup>。 </p> <div class="mw-heading mw-heading2"><h2 id="概念詮釋"><span id=".E6.A6.82.E5.BF.B5.E8.A9.AE.E9.87.8B"></span>概念詮釋</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%A6%82%E7%8E%87%E8%AB%96&action=edit&section=16" title="編輯小節: 概念詮釋"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote">内文：<a href="/wiki/%E6%A6%82%E7%8E%87%E5%98%85%E8%A9%AE%E9%87%8B" title="概率嘅詮釋">概率嘅詮釋</a></div> <p>廿一世紀初嘅概率論係一個廣受人認同嘅<a href="/wiki/%E6%95%B8%E5%AD%B8%E7%90%86%E8%AB%96" title="數學理論">數學理論</a>：<a href="/wiki/%E6%95%B8%E5%AD%B8" title="數學">數學</a>上對概率嘅分析始於 8 至 13 世紀（<a href="/wiki/%E4%BC%8A%E6%96%AF%E8%98%AD%E9%BB%83%E9%87%91%E6%99%82%E4%BB%A3" title="伊斯蘭黃金時代">伊斯蘭黃金時代</a>）期間，當時嘅<a href="/wiki/%E9%98%BF%E6%8B%89%E4%BC%AF" title="阿拉伯">阿拉伯</a><a href="/wiki/%E6%95%B8%E5%AD%B8%E5%AE%B6" title="數學家">數學家</a>喺度研究<a href="/wiki/%E5%AF%86%E7%A2%BC%E5%AD%B8" title="密碼學">密碼學</a>，有諗到例如「點樣令到啲<a href="/wiki/%E5%AF%86%E6%96%87" title="密文">密文</a>望落似完全<a href="/wiki/%E9%9A%A8%E6%A9%9F" title="隨機">隨機</a>（難以確定噉知道）」等嘅問題<sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup>；打後嘅 17 世紀數學家郁手分析<a href="/wiki/%E6%93%B2%E9%AA%B0%E4%BB%94" class="mw-redirect" title="擲骰仔">擲骰仔</a>等嘅<a href="/wiki/%E6%A9%9F%E7%8E%87%E9%81%8A%E6%88%B2" title="機率遊戲">機率遊戲</a>，令到概率論嘅諗法萌芽；到咗廿世紀上半橛，<a href="/wiki/%E8%98%87%E8%81%AF" title="蘇聯">蘇聯</a>數學家<a href="/wiki/%E5%AE%89%E5%BE%B7%E9%9B%B7%C2%B7%E6%9F%AF%E7%88%BE%E8%8E%AB%E5%93%A5%E6%B4%9B%E5%A4%AB" title="安德雷·柯爾莫哥洛夫">安德雷·柯爾莫哥洛夫</a><sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>英 25<span class="cite-bracket">]</span></a></sup>將概率同相關概念整合做一套<a href="/wiki/%E5%BD%A2%E5%BC%8F%E5%8C%96" class="mw-redirect" title="形式化">形式化</a>嘅理論，令到概率論正式成為一個嚴謹嘅數學領域，並且俾人廣泛噉應用喺<a href="/wiki/%E7%B5%B1%E8%A8%88" class="mw-redirect" title="統計">統計</a>同<a href="/wiki/%E8%B3%87%E8%A8%8A%E7%A7%91%E6%8A%80" title="資訊科技">資訊科技</a>等嘅領域上<sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup>。 </p> <div class="mw-heading mw-heading3"><h3 id="古典定義"><span id=".E5.8F.A4.E5.85.B8.E5.AE.9A.E7.BE.A9"></span>古典定義</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%A6%82%E7%8E%87%E8%AB%96&action=edit&section=17" title="編輯小節: 古典定義"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote">内文：<a href="/wiki/%E5%8F%A4%E5%85%B8%E6%A6%82%E7%8E%87%E5%AE%9A%E7%BE%A9" title="古典概率定義">古典概率定義</a></div> <p>雖然概率論咁成功，但<a href="/wiki/%E5%93%B2%E5%AD%B8" title="哲學">哲學</a>同<a href="/wiki/%E7%89%A9%E7%90%86%E5%AD%B8" title="物理學">物理學</a>等領域嘅研究者一路都有喺度諗「概率到底代表緊啲乜」嘅問題。18 世紀<a href="/wiki/%E6%B3%95%E5%9C%8B" title="法國">法國</a>數學家<a href="/wiki/%E6%8B%89%E6%99%AE%E6%8B%89%E6%96%AF" title="拉普拉斯">拉普拉斯</a><sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>英 26<span class="cite-bracket">]</span></a></sup>係噉樣<a href="/wiki/%E5%AE%9A%E7%BE%A9" title="定義">定義</a><b>概率</b>嘅<sup id="cite_ref-61" class="reference"><a href="#cite_note-61"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-62" class="reference"><a href="#cite_note-62"><span class="cite-bracket">[</span>註 5<span class="cite-bracket">]</span></a></sup>：如果 </p> <ul><li>一場<a href="/wiki/%E9%9A%A8%E6%A9%9F%E5%AF%A6%E9%A9%97" class="mw-redirect" title="隨機實驗">隨機實驗</a>可以引致 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> 個<a href="/wiki/%E4%BA%92%E6%96%A5" title="互斥">互斥</a>同一樣咁有可能嘅<a href="/wiki/%E5%8F%AF%E8%83%BD%E7%B5%90%E6%9E%9C" title="可能結果">結果</a>，而且</li> <li>呢 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> 個結果當中有 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa9233ef6a6a2c0576dbf65490ddb8307fde494" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.331ex; height:2.509ex;" alt="{\displaystyle N_{A}}"></span> 咁多個會涉及<a href="/wiki/%E4%BA%8B%E4%BB%B6_(%E6%A6%82%E7%8E%87%E8%AB%96)" title="事件 (概率論)">事件</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> 發生；</li></ul> <p>噉 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> 嘅概率（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f264d19e21604793c6dc54f8044df454db82744" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.298ex; height:2.843ex;" alt="{\displaystyle P(A)}"></span>）就可以定義做 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A)={N_{A} \over N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mi>N</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(A)={N_{A} \over N}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b3617b454b643977609e21725629d9581ef6e18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.563ex; height:5.343ex;" alt="{\displaystyle P(A)={N_{A} \over N}}"></span></dd></dl> <p>呢個定義俾好多學者覺得有缺憾：首先，個定義淨係可以用喺可能結果嘅數量有限嘅情況，但某啲重要嘅隨機實驗（例如係「一路<a href="/wiki/%E6%8E%9F%E9%8A%80%E4%BB%94" title="掟銀仔">掟銀仔</a>，掟到出公為止」）理論上有無限咁多個可能結果；除此之外又有人指，古典定義有<a href="/wiki/%E5%BE%AA%E7%92%B0%E9%82%8F%E8%BC%AF" class="mw-redirect" title="循環邏輯">循環邏輯</a><sup id="cite_ref-63" class="reference"><a href="#cite_note-63"><span class="cite-bracket">[</span>英 27<span class="cite-bracket">]</span></a></sup>嘅問題－「一個冇出千嘅銀仔出公出字嘅概率係咁多咁多」，但同時一個冇出千嘅銀仔定義上就係「出公出字機會一樣」嘅<sup id="cite_ref-64" class="reference"><a href="#cite_note-64"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup>。因為噉，打後嘅數學界又出咗 </p> <ul><li><a href="/wiki/%E9%A0%BB%E7%8E%87%E5%AD%B8%E6%B4%BE%E6%A6%82%E7%8E%87" title="頻率學派概率">頻率派</a><sup id="cite_ref-65" class="reference"><a href="#cite_note-65"><span class="cite-bracket">[</span>英 28<span class="cite-bracket">]</span></a></sup>，一件事件嘅概率係「件事件經過多次實驗之後嘅相對頻率嘅<a href="/wiki/%E6%A5%B5%E9%99%90_(%E6%95%B8%E5%AD%B8)" title="極限 (數學)">極限</a>」</li> <li><a href="/wiki/%E8%B2%9D%E8%91%89%E6%96%AF%E6%A6%82%E7%8E%87" title="貝葉斯概率">貝葉斯派</a><sup id="cite_ref-66" class="reference"><a href="#cite_note-66"><span class="cite-bracket">[</span>英 29<span class="cite-bracket">]</span></a></sup>，一件事件嘅概率係「個觀察者根據過往經驗，覺得有幾預期件事件會發生」</li></ul> <p>等嘅諗法<sup id="cite_ref-67" class="reference"><a href="#cite_note-67"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup>。 </p> <div class="mw-heading mw-heading3"><h3 id="物理思考"><span id=".E7.89.A9.E7.90.86.E6.80.9D.E8.80.83"></span>物理思考</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%A6%82%E7%8E%87%E8%AB%96&action=edit&section=18" title="編輯小節: 物理思考"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote">睇埋：<a href="/wiki/%E6%B1%BA%E5%AE%9A%E8%AB%96" title="決定論">決定論</a>同<a href="/wiki/%E9%87%8F%E5%AD%90%E5%8A%9B%E5%AD%B8" title="量子力學">量子力學</a></div> <p>有關概率嘅本質嘅問題仲涉及物理學同<a href="/wiki/%E5%AE%87%E5%AE%99%E5%AD%B8" title="宇宙學">宇宙學</a>上嘅思考。喺廿世紀打前嘅科學家好多時都抱持<a href="/wiki/%E6%B1%BA%E5%AE%9A%E8%AB%96" title="決定論">決定論</a><sup id="cite_ref-det_3-1" class="reference"><a href="#cite_note-det-3"><span class="cite-bracket">[</span>英 1<span class="cite-bracket">]</span></a></sup>嘅觀念：喺最基本上，決定論係一種世界觀，認為<a href="/wiki/%E5%AE%87%E5%AE%99" title="宇宙">宇宙</a>裏面嘅每一件事件都係由打前嘅事件（<a href="/wiki/%E5%8E%9F%E5%9B%A0" class="mw-redirect" title="原因">原因</a>）決定嘅；根據呢種睇法，概率呢個概念之所以存在，純粹係反映咗觀察者<a href="/wiki/%E8%B3%87%E8%A8%8A" title="資訊">資訊</a>上嘅不足－「如果一個觀察者完美噉知嗮嗰一刻『每粒<a href="/wiki/%E5%8E%9F%E5%AD%90" title="原子">原子</a>喺邊』同『每粒原子以乜嘢<a href="/wiki/%E9%80%9F%E5%BA%A6" title="速度">速度</a>移動』等嘅資訊，佢將會有能力按<a href="/wiki/%E7%89%A9%E7%90%86%E5%AE%9A%E5%BE%8B" title="物理定律">物理定律</a>完美噉預測個銀仔會係公定字，但喺實際情況當中，人唔能夠攞到嗮呢啲資訊，所以淨係有得估呢啲事件發生嘅概率」。詳情可以睇吓<a href="/wiki/%E6%8B%89%E6%99%AE%E6%8B%89%E6%96%AF%E9%AD%94" title="拉普拉斯魔">拉普拉斯魔</a><sup id="cite_ref-68" class="reference"><a href="#cite_note-68"><span class="cite-bracket">[</span>英 30<span class="cite-bracket">]</span></a></sup>相關嘅概念<sup id="cite_ref-laplacedemon_4-1" class="reference"><a href="#cite_note-laplacedemon-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>。 </p> <figure class="mw-halign-center" typeof="mw:File/Thumb"><span><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Wave-particle_duality.ogv/510px--Wave-particle_duality.ogv.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="510" height="287" data-durationhint="117" data-mwtitle="Wave-particle_duality.ogv" data-mwprovider="wikimediacommons" resource="/wiki/File:Wave-particle_duality.ogv"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/e/e4/Wave-particle_duality.ogv/Wave-particle_duality.ogv.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="854" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/e/e4/Wave-particle_duality.ogv/Wave-particle_duality.ogv.720p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="720p.vp9.webm" data-width="1280" data-height="720" /><source src="//upload.wikimedia.org/wikipedia/commons/e/e4/Wave-particle_duality.ogv" type="video/ogg; codecs="theora, vorbis"" data-width="1280" data-height="720" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/e/e4/Wave-particle_duality.ogv/Wave-particle_duality.ogv.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="426" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/e/e4/Wave-particle_duality.ogv/Wave-particle_duality.ogv.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="640" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/e/e4/Wave-particle_duality.ogv/Wave-particle_duality.ogv.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="640" data-height="360" /></video></span><figcaption>一段有關<a href="/wiki/%E6%B3%A2%E7%B2%92%E4%BA%8C%E8%B1%A1%E6%80%A7" class="mw-redirect" title="波粒二象性">波粒二象性</a>嘅英文動畫；一般<a href="/wiki/%E5%8F%A4%E5%85%B8%E7%89%A9%E7%90%86%E5%AD%B8" title="古典物理學">古典物理學</a>嘅諗法係<a href="/wiki/%E6%B3%A2%E5%8B%95" title="波動">波動</a>同<a href="/wiki/%E7%B2%92%E5%AD%90" title="粒子">粒子</a>係兩樣唔同嘅嘢，但廿世紀頭半橛嘅物理實驗發現粒子有時會有好似波動噉嘅行為，例如係<a href="/wiki/%E9%9B%BB%E5%AD%90" title="電子">電子</a>嘅<a href="/wiki/%E7%B9%9E%E5%B0%84" title="繞射">繞射</a>現象。</figcaption></figure> <p>不過，古典嘅決定論喺廿世紀受到挑戰：廿世紀上半橛係<a href="/wiki/%E9%87%8F%E5%AD%90%E5%8A%9B%E5%AD%B8" title="量子力學">量子力學</a>崛起嘅時期，呢啲研究掂到<a href="/wiki/%E6%B3%A2%E7%B2%92%E4%BA%8C%E8%B1%A1%E6%80%A7" class="mw-redirect" title="波粒二象性">波粒二象性</a><sup id="cite_ref-69" class="reference"><a href="#cite_note-69"><span class="cite-bracket">[</span>英 31<span class="cite-bracket">]</span></a></sup>同<a href="/wiki/%E5%93%A5%E6%9C%AC%E5%93%88%E6%A0%B9%E8%A9%AE%E9%87%8B" title="哥本哈根詮釋">哥本哈根詮釋</a>等嘅議題；簡單講，量子力學發現微觀<a href="/wiki/%E7%B2%92%E5%AD%90" title="粒子">粒子</a>同時會有<a href="/wiki/%E6%B3%A2%E5%8B%95" title="波動">波動</a>（<a href="/wiki/%E8%83%BD%E9%87%8F" title="能量">能量</a>嘅擾動）同粒子（一種<a href="/wiki/%E7%89%A9%E8%B3%AA" title="物質">物質</a>）嘅特性。例如係<a href="/wiki/%E5%85%89" title="光">光</a>喺古典物理學當中俾人當係波動一種，而唔係物質，但量子力學就話光有某啲粒子先至有嘅特性；根據哥本哈根詮釋，一粒粒子嘅「波動」係表示緊嗰粒粒子喺唔同位置嘅概率－喺做<a href="/wiki/%E9%87%8F%E5%BA%A6" title="量度">量度</a>之前，粒粒子會喺空間入面有個波動，每個位置嘅波動大細表示「嗰粒粒子喺嗰個位嘅概率」，而當有觀察者郁手做量度嗰陣，粒粒子會<a href="/wiki/%E6%B3%A2%E5%87%BD%E6%95%B8%E5%B4%A9%E5%A1%8C" title="波函數崩塌">即刻出現喺其中一個位置</a>。呢個諗法引致咗一個問題：如果呢個詮釋係啱嘅，噉即係話宇宙入面至少有一啲現象喺本質上係隨機嘅；呢個諗法令到古典嘅決定論大受打擊，亦都表示概率唔淨只係反映人嘅知識不足，而係宇宙本質上有嘅一種特性<sup id="cite_ref-70" class="reference"><a href="#cite_note-70"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-71" class="reference"><a href="#cite_note-71"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup>。 </p> <div class="mw-heading mw-heading2"><h2 id="統計應用"><span id=".E7.B5.B1.E8.A8.88.E6.87.89.E7.94.A8"></span>統計應用</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%A6%82%E7%8E%87%E8%AB%96&action=edit&section=19" title="編輯小節: 統計應用"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Simple_random_sampling_canto_%26_mando.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Simple_random_sampling_canto_%26_mando.png/300px-Simple_random_sampling_canto_%26_mando.png" decoding="async" width="300" height="231" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Simple_random_sampling_canto_%26_mando.png/450px-Simple_random_sampling_canto_%26_mando.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Simple_random_sampling_canto_%26_mando.png/600px-Simple_random_sampling_canto_%26_mando.png 2x" data-file-width="618" data-file-height="476" /></a><figcaption>一個<a href="/wiki/%E6%8A%BD%E6%A8%A3" title="抽樣">抽樣</a>嘅過程係由一大柞研究對象（總體）嗰度抽一部份（樣本）出嚟研究－要睇嗮所有研究對象太嘥時間或者太嘥錢所以唔可行。</figcaption></figure> <div role="note" class="hatnote">睇埋：<a href="/wiki/%E7%B5%B1%E8%A8%88%E5%AD%B8" title="統計學">統計學</a>、<a href="/wiki/%E6%A9%9F%E6%A2%B0%E5%AD%B8%E7%BF%92" title="機械學習">機械學習</a>同<a href="/wiki/%E8%B3%87%E8%A8%8A%E7%90%86%E8%AB%96" title="資訊理論">資訊理論</a></div> <p>機會率係<a href="/wiki/%E7%B5%B1%E8%A8%88%E5%AD%B8" title="統計學">統計學</a><sup id="cite_ref-72" class="reference"><a href="#cite_note-72"><span class="cite-bracket">[</span>英 32<span class="cite-bracket">]</span></a></sup>同相關領域上實要諗嘅課題：統計學嘅重要一環係分析<a href="/wiki/%E7%A7%91%E5%AD%B8%E6%96%B9%E6%B3%95" title="科學方法">科學方法</a><sup id="cite_ref-73" class="reference"><a href="#cite_note-73"><span class="cite-bracket">[</span>英 33<span class="cite-bracket">]</span></a></sup>上得到嘅<a href="/wiki/%E6%95%B8%E6%93%9A" title="數據">數據</a>。 </p><p>科學方法本質上就涉及研究者由一個<a href="/wiki/%E7%B8%BD%E9%AB%94" title="總體">總體</a><sup id="cite_ref-74" class="reference"><a href="#cite_note-74"><span class="cite-bracket">[</span>英 34<span class="cite-bracket">]</span></a></sup>入面攞一個<a href="/wiki/%E6%A8%A3%E6%9C%AC" title="樣本">樣本</a><sup id="cite_ref-75" class="reference"><a href="#cite_note-75"><span class="cite-bracket">[</span>英 35<span class="cite-bracket">]</span></a></sup>出嚟，並且嘗試靠分析手上嘅樣本嚟增進自己對個總體嘅認識；呢種做法本質上就有<a href="/wiki/%E4%B8%8D%E7%A2%BA%E5%AE%9A" title="不確定">不確定</a>－理論上，研究者永遠都唔能夠肯定，個樣本實係代表得到個總體，例如研究者想研究<a href="/wiki/%E7%8B%BC" title="狼">狼</a>嘅<a href="/wiki/%E9%87%8D%E9%87%8F" title="重量">體重</a>，因為人力物力嘅限制，佢冇可能研究嗮古往今來所有嘅狼，於是佢就去搵 100 隻狼嚟做樣本研究，佢量度到呢個樣本嘅狼平均體重係 40 <a href="/wiki/%E5%85%AC%E6%96%A4" class="mw-redirect" title="公斤">kg</a>，不過就最嚴格嘅<a href="/wiki/%E9%82%8F%E8%BC%AF" title="邏輯">邏輯</a>基準嚟講，呢個數可能真係代表到全世界嘅狼，但又有可能全世界嘅狼嘅平均體重查實係 60 公斤，個研究者之所以搵到 40 公斤呢個數只係佢咁啱得咁橋唔好彩，抽到個代表唔到個總體嘅樣本－<a href="/wiki/%E9%9A%A8%E6%A9%9F" title="隨機">隨機</a>係統計分析上無可避免嘅一部份<sup id="cite_ref-feller1968_1-3" class="reference"><a href="#cite_note-feller1968-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-76" class="reference"><a href="#cite_note-76"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup>。 </p> <blockquote class="toccolours" style="float: none; padding: 0.3em 1em;"> <p><b>例：信心區間</b><br /> </p><p>好多統計學上會用嘅分析概念都係以概率做基礎嘅。例如係<a href="/wiki/%E4%BF%A1%E5%BF%83%E5%8D%80%E9%96%93" title="信心區間">信心區間</a><sup id="cite_ref-77" class="reference"><a href="#cite_note-77"><span class="cite-bracket">[</span>英 36<span class="cite-bracket">]</span></a></sup>噉：研究人員可以睇到嘅就淨係得個樣本啲數值，而個總體嗰柞真實數值原則上係不可知嘅；信心區間就係指「有信心總體個真實數值係喺入面嘅區間」，喺做統計嗰時會俾人攞嚟表述個樣本嘅數值同個總體嘅真實數值之間估計差幾遠， </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(L_{n}<\theta <U_{n})=\gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo><</mo> <mi>θ</mi> <mo><</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(L_{n}<\theta <U_{n})=\gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6246d6f24638c7cd2f1d95646e387fea5c926801" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.81ex; height:2.843ex;" alt="{\displaystyle P(L_{n}<\theta <U_{n})=\gamma }"></span></dd></dl> <p>舉個例說明，最常用嘅係「95% 信心區間」（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma =0.95}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mn>0.95</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma =0.95}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da2269657267c9cbd5a29b9c114cc666dd2271b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.495ex; height:2.676ex;" alt="{\displaystyle \gamma =0.95}"></span>），啲科研人員會用個樣本入面嘅海豚嘅身長平均值嚟估計嗰個不可知嘅（例如）「世上所有狼嘅身長嘅平均值」（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span>），而佢哋可以用一啲統計方法計個「狼身長平均值嘅 95% 信心區間」出嚟－呢個值係指「有信心 95% 機會世上所有狼嘅身長嘅平均值嘅真實數值係喺 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebec334cb04f246db1139e2ca6be0b957d2ef520" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.801ex; height:2.509ex;" alt="{\displaystyle L_{n}}"></span> 同 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67b641ce10cb5e865397c4efe647cbffed98a024" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.806ex; height:2.509ex;" alt="{\displaystyle U_{n}}"></span> 之間」－信心區間正正就係以概率嘅形式嚟表達嘅<sup id="cite_ref-78" class="reference"><a href="#cite_note-78"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup>。 </p> </blockquote> <div class="mw-heading mw-heading2"><h2 id="睇埋"><span id=".E7.9D.87.E5.9F.8B"></span>睇埋</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%A6%82%E7%8E%87%E8%AB%96&action=edit&section=20" title="編輯小節: 睇埋"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E6%A9%9F%E7%8E%87%E9%81%8A%E6%88%B2" title="機率遊戲">機率遊戲</a>同埋<a href="/wiki/%E8%B3%AD%E9%8C%A2" class="mw-redirect" title="賭錢">賭錢</a></li> <li><a href="/wiki/%E9%9A%A8%E6%A9%9F%E6%95%B8%E7%94%9F%E6%88%90" title="隨機數生成">RNG</a> 同<a href="/wiki/%E6%93%AC%E4%BA%82%E6%95%B8%E7%94%A2%E7%94%9F" title="擬亂數產生">擬亂數產生</a></li> <li><a href="/wiki/%E5%BF%AB%E6%80%9D%E9%82%8F%E8%BC%AF" title="快思邏輯">快思邏輯</a></li> <li><a href="/wiki/%E7%B5%B1%E8%A8%88%E5%8A%9B%E5%AD%B8" title="統計力學">統計力學</a></li> <li><a href="/wiki/%E5%AF%86%E7%A2%BC%E5%AD%B8" title="密碼學">密碼學</a></li> <li><a href="/wiki/%E7%9F%A5%E8%AD%98%E8%AB%96" title="知識論">知識論</a></li></ul> <div style="clear: both;"></div> <div class="mw-heading mw-heading2"><h2 id="文獻"><span id=".E6.96.87.E7.8D.BB"></span>文獻</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%A6%82%E7%8E%87%E8%AB%96&action=edit&section=21" title="編輯小節: 文獻"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="div-col columns column-count column-count-2" style="-moz-column-count: 2; -webkit-column-count: 2; column-count: 2; font-size:90%; column-count:1"> <ul><li>Billingsley, P. (2008). <i>Probability and measure</i>. John Wiley & Sons.</li> <li>Gut, A. (2013). <i>Probability: a graduate course</i> (Vol. 75). Springer Science & Business Media.</li> <li>Kallenberg, O. (2006). <i>Foundations of modern probability</i>. Springer Science & Business Media.</li> <li>Kallenberg, O. (2006). <i>Probabilistic symmetries and invariance principles</i>. Springer Science & Business Media.</li> <li>Tijms, H. (2012). <i>Understanding probability</i>. Cambridge University Press.</li> <li>Tucker, H. G. (2013). <i>A graduate course in probability</i>. Courier Corporation.</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="註釋"><span id=".E8.A8.BB.E9.87.8B"></span>註釋</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%A6%82%E7%8E%87%E8%AB%96&action=edit&section=22" title="編輯小節: 註釋"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r2020813">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width reflist-columns-2"> <ol class="references"> <li id="cite_note-11"><span class="mw-cite-backlink"><a href="#cite_ref-11">↑</a></span> <span class="reference-text">概率論上所講嘅「<a href="/wiki/%E9%9A%A8%E6%A9%9F%E5%AF%A6%E9%A9%97" class="mw-redirect" title="隨機實驗">實驗</a>」同一般<a href="/wiki/%E7%A7%91%E5%AD%B8%E6%96%B9%E6%B3%95" title="科學方法">科學方法</a>上講嘅「<a href="/wiki/%E5%AF%A6%E9%A9%97" title="實驗">實驗</a>」係兩個唔同嘅概念。</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><a href="#cite_ref-16">↑</a></span> <span class="reference-text">嚴格噉講，機率係 0 表示嗰件事件發生嘅機會非常之咁細，好似<a href="/wiki/%E7%84%A1%E7%AA%AE%E5%B0%8F%E9%87%8F" title="無窮小量">無窮小量</a>噉，但唔係完全冇可能發生。機率係 1 可以用同樣道理想像。</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><a href="#cite_ref-22">↑</a></span> <span class="reference-text">不交集簡單講就係冇可能同時發生嘅事件。例如家陣擲三粒骰仔：<br />「掟到 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,3,5\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>5</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,3,5\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dff31162647fd4911bca0b34f555de99928d1cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.88ex; height:2.843ex;" alt="{\displaystyle \{1,3,5\}}"></span>」同「掟到 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{2,4,6\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>6</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{2,4,6\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc24923d36d36829caf41d4e801eeb5409bb0766" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.88ex; height:2.843ex;" alt="{\displaystyle \{2,4,6\}}"></span>」係冇可能同時發生嘅；但<br />「掟到最少一個 2」同「掟到最少一個 4」就係有可能同時發生嘅。</span> </li> <li id="cite_note-56"><span class="mw-cite-backlink"><a href="#cite_ref-56">↑</a></span> <span class="reference-text"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> 係二項分佈當中有嘅一個<a href="/wiki/%E5%8F%83%E6%95%B8" title="參數">參數</a>。</span> </li> <li id="cite_note-62"><span class="mw-cite-backlink"><a href="#cite_ref-62">↑</a></span> <span class="reference-text">呢個就係所謂嘅<a href="/wiki/%E5%8F%A4%E5%85%B8%E6%A6%82%E7%8E%87%E5%AE%9A%E7%BE%A9" title="古典概率定義">古典定義</a>。</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="詞彙"><span id=".E8.A9.9E.E5.BD.99"></span>詞彙</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%A6%82%E7%8E%87%E8%AB%96&action=edit&section=23" title="編輯小節: 詞彙"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2020813"><div class="reflist reflist-columns references-column-width reflist-columns-2"> <ol class="references"> <li id="cite_note-det-3"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-det_3-0">1.0</a></sup> <sup><a href="#cite_ref-det_3-1">1.1</a></sup></span> <span class="reference-text">determinism</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><a href="#cite_ref-10">↑</a></span> <span class="reference-text">uncertainty</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><a href="#cite_ref-12">↑</a></span> <span class="reference-text">sample space</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><a href="#cite_ref-15">↑</a></span> <span class="reference-text">probability</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><a href="#cite_ref-19">↑</a></span> <span class="reference-text">probability axioms</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><a href="#cite_ref-21">↑</a></span> <span class="reference-text">elementary event</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><a href="#cite_ref-23">↑</a></span> <span class="reference-text">intersection</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><a href="#cite_ref-24">↑</a></span> <span class="reference-text">union</span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><a href="#cite_ref-25">↑</a></span> <span class="reference-text">Venn diagram</span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><a href="#cite_ref-27">↑</a></span> <span class="reference-text">complementary event</span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><a href="#cite_ref-29">↑</a></span> <span class="reference-text">mutually exclusive events</span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><a href="#cite_ref-31">↑</a></span> <span class="reference-text">non-mutually exclusive events</span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><a href="#cite_ref-32">↑</a></span> <span class="reference-text">conditional probability</span> </li> <li id="cite_note-34"><span class="mw-cite-backlink"><a href="#cite_ref-34">↑</a></span> <span class="reference-text">statistical independence</span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><a href="#cite_ref-37">↑</a></span> <span class="reference-text">conditional independence</span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><a href="#cite_ref-38">↑</a></span> <span class="reference-text">conditionally independent given C</span> </li> <li id="cite_note-40"><span class="mw-cite-backlink"><a href="#cite_ref-40">↑</a></span> <span class="reference-text">probability distribution</span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><a href="#cite_ref-42">↑</a></span> <span class="reference-text">Bernoulli distribution</span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><a href="#cite_ref-44">↑</a></span> <span class="reference-text">normal distribution</span> </li> <li id="cite_note-46"><span class="mw-cite-backlink"><a href="#cite_ref-46">↑</a></span> <span class="reference-text">convergence of random variable</span> </li> <li id="cite_note-47"><span class="mw-cite-backlink"><a href="#cite_ref-47">↑</a></span> <span class="reference-text">limit</span> </li> <li id="cite_note-49"><span class="mw-cite-backlink"><a href="#cite_ref-49">↑</a></span> <span class="reference-text">law of large numbers，LLN</span> </li> <li id="cite_note-51"><span class="mw-cite-backlink"><a href="#cite_ref-51">↑</a></span> <span class="reference-text">sample average / observed average</span> </li> <li id="cite_note-53"><span class="mw-cite-backlink"><a href="#cite_ref-53">↑</a></span> <span class="reference-text">central limit theorem，CLT</span> </li> <li id="cite_note-58"><span class="mw-cite-backlink"><a href="#cite_ref-58">↑</a></span> <span class="reference-text">Andrey Kolmogorov</span> </li> <li id="cite_note-60"><span class="mw-cite-backlink"><a href="#cite_ref-60">↑</a></span> <span class="reference-text">Pierre-Simon Laplace</span> </li> <li id="cite_note-63"><span class="mw-cite-backlink"><a href="#cite_ref-63">↑</a></span> <span class="reference-text">circular logic</span> </li> <li id="cite_note-65"><span class="mw-cite-backlink"><a href="#cite_ref-65">↑</a></span> <span class="reference-text">frequentist</span> </li> <li id="cite_note-66"><span class="mw-cite-backlink"><a href="#cite_ref-66">↑</a></span> <span class="reference-text">Bayesian</span> </li> <li id="cite_note-68"><span class="mw-cite-backlink"><a href="#cite_ref-68">↑</a></span> <span class="reference-text">Laplace's demon</span> </li> <li id="cite_note-69"><span class="mw-cite-backlink"><a href="#cite_ref-69">↑</a></span> <span class="reference-text">wave-particle duality</span> </li> <li id="cite_note-72"><span class="mw-cite-backlink"><a href="#cite_ref-72">↑</a></span> <span class="reference-text">statistics</span> </li> <li id="cite_note-73"><span class="mw-cite-backlink"><a href="#cite_ref-73">↑</a></span> <span class="reference-text">scientific method</span> </li> <li id="cite_note-74"><span class="mw-cite-backlink"><a href="#cite_ref-74">↑</a></span> <span class="reference-text">population</span> </li> <li id="cite_note-75"><span class="mw-cite-backlink"><a href="#cite_ref-75">↑</a></span> <span class="reference-text">sample</span> </li> <li id="cite_note-77"><span class="mw-cite-backlink"><a href="#cite_ref-77">↑</a></span> <span class="reference-text">confidence interval，CI</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="引咗"><span id=".E5.BC.95.E5.92.97"></span>引咗</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%A6%82%E7%8E%87%E8%AB%96&action=edit&section=24" title="編輯小節: 引咗"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2020813"><div class="reflist reflist-columns references-column-width reflist-columns-2"> <ol class="references"> <li id="cite_note-feller1968-1"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-feller1968_1-0">1.0</a></sup> <sup><a href="#cite_ref-feller1968_1-1">1.1</a></sup> <sup><a href="#cite_ref-feller1968_1-2">1.2</a></sup> <sup><a href="#cite_ref-feller1968_1-3">1.3</a></sup></span> <span class="reference-text">William Feller, <i>An Introduction to Probability Theory and Its Applications</i>, (Vol 1), 3rd Ed, (1968), Wiley.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Kallenberg, O. (2006). <i>Foundations of modern probability</i>. Springer Science & Business Media.</span> </li> <li id="cite_note-laplacedemon-4"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-laplacedemon_4-0">3.0</a></sup> <sup><a href="#cite_ref-laplacedemon_4-1">3.1</a></sup></span> <span class="reference-text">Richard Langdon Franklin (1968). <i>Freewill and determinism: a study of rival conceptions of man</i>. Routledge & K. Paul.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">Laplace, Pierre Simon. <i>A Philosophical Essay on Probabilities</i>, translated into English from the original French 6th ed. by Truscott, F.W. and Emory, F.L., Dover Publications (New York, 1951).</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text">Moore, W.J. (1992). <i>Schrödinger: Life and Thought</i>. Cambridge University Press. p. 479.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text">Stephen Hawking's Grand Design (2010), page 32: "the molecular basis of biology shows that biological processes are governed by the laws of physics and chemistry and therefore are as determined as the orbits of the planets...so it seems that we are no more than biological machines and that free will is just an illusion", and page 72: "Quantum physics might seem to undermine the idea that nature is governed by laws, but that is not the case. Instead it leads us to accept a new form of determinism: Given the state of a system at some time, the laws of nature determine the probabilities of various futures and pasts rather than determining the future and past with certainty." (discussing a Many worlds interpretation).</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><a href="#cite_ref-8">↑</a></span> <span class="reference-text">Mohri, Mehryar; Rostamizadeh, Afshin; Talwalkar, Ameet (2012). <i>Foundations of Machine Learning</i>. USA, Massachusetts: MIT Press.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><a href="#cite_ref-9">↑</a></span> <span class="reference-text">Dervishi, Kay (2019-06-18). "<a rel="nofollow" class="external text" href="https://www.cityandstateny.com/articles/policy/gambling/new-york-games-chance-and-skill.html">Other games of chance and skill on Albany's agenda</a> <a href="/wiki/Wayback_Machine" title="Wayback Machine">互聯網檔案館</a>嘅<a rel="nofollow" class="external text" href="https://web.archive.org/web/20210518083130/https://www.cityandstateny.com/articles/policy/gambling/new-york-games-chance-and-skill.html">歸檔</a>，歸檔日期2021年5月18號，.". <i>CSNY</i>.</span> </li> <li id="cite_note-ross2010-13"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-ross2010_13-0">9.0</a></sup> <sup><a href="#cite_ref-ross2010_13-1">9.1</a></sup></span> <span class="reference-text">Ross, Sheldon (2010). <i>A First Course in Probability</i> (8th ed.). Pearson Prentice Hall. pp. 26–27.</span> </li> <li id="cite_note-pap1984-14"><span class="mw-cite-backlink"><a href="#cite_ref-pap1984_14-0">↑</a></span> <span class="reference-text">Papoulis, A. (1984). "Bernoulli Trials". <i>Probability, Random Variables, and Stochastic Processes</i> (2nd ed.). New York: McGraw-Hill. pp. 57–63.</span> </li> <li id="cite_note-bain1992-17"><span class="mw-cite-backlink"><a href="#cite_ref-bain1992_17-0">↑</a></span> <span class="reference-text">Bain, Lee J.; Engelhardt, Max (1992). <i>Introduction to Probability and Mathematical Statistics</i> (2nd ed.). Belmont, California: Brooks/Cole. p. 53.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><a href="#cite_ref-18">↑</a></span> <span class="reference-text">Murphy, K. P. (2012). <i>Machine learning: a probabilistic perspective</i>, p. 35. MIT press.</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><a href="#cite_ref-20">↑</a></span> <span class="reference-text">Kolmogorov, Andrey (1950) [1933]. <i>Foundations of the theory of probability</i>. New York, USA: Chelsea Publishing Company.</span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><a href="#cite_ref-26">↑</a></span> <span class="reference-text">Mahmoodian, Ebadollah S.; Rezaie, M.; Vatan, F. (March 1987). "Generalization of Venn Diagram". <i>Eighteenth Annual Iranian Mathematics Conference</i>. Tehran and Isfahan, Iran.</span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><a href="#cite_ref-28">↑</a></span> <span class="reference-text">Robert R. Johnson, Patricia J. Kuby: <i>Elementary Statistics</i>. Cengage Learning 2007, <a href="/wiki/Special:%E6%9B%B8%E6%9C%AC%E4%BE%86%E6%BA%90/9780495383864" class="internal mw-magiclink-isbn">ISBN 978-0-495-38386-4</a>, p. 229.</span> </li> <li id="cite_note-miller2012-30"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-miller2012_30-0">16.0</a></sup> <sup><a href="#cite_ref-miller2012_30-1">16.1</a></sup> <sup><a href="#cite_ref-miller2012_30-2">16.2</a></sup></span> <span class="reference-text">Miller, Scott; Childers, Donald (2012). <i>Probability and Random Processes</i> (Second ed.). Academic Press. p. 8. <a href="/wiki/Special:%E6%9B%B8%E6%9C%AC%E4%BE%86%E6%BA%90/9780123869814" class="internal mw-magiclink-isbn">ISBN 978-0-12-386981-4</a>. The sample space is the collection or set of 'all possible' distinct (collectively exhaustive and mutually exclusive) outcomes of an experiment."</span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><a href="#cite_ref-33">↑</a></span> <span class="reference-text">Olofsson (2005) p. 29.</span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><a href="#cite_ref-35">↑</a></span> <span class="reference-text">Olofsson (2005) p. 35.</span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><a href="#cite_ref-36">↑</a></span> <span class="reference-text">Russell, Stuart; Norvig, Peter (2002). <i>Artificial Intelligence: A Modern Approach</i>. Prentice Hall. p. 478.</span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><a href="#cite_ref-39">↑</a></span> <span class="reference-text">Dawid, A. P. (1979). "Conditional Independence in Statistical Theory". <i>Journal of the Royal Statistical Society, Series B</i>. 41 (1): 1–31.</span> </li> <li id="cite_note-robert2008-41"><span class="mw-cite-backlink"><a href="#cite_ref-robert2008_41-0">↑</a></span> <span class="reference-text">Ash, Robert B. (2008). <i>Basic probability theory</i> (Dover ed.). Mineola, N.Y.: Dover Publications. pp. 66–69.</span> </li> <li id="cite_note-bertsekas2002-43"><span class="mw-cite-backlink"><a href="#cite_ref-bertsekas2002_43-0">↑</a></span> <span class="reference-text">Bertsekas, Dimitri P. (2002). <i>Introduction to Probability</i>. Tsitsiklis, John N., Τσιτσικλής, Γιάννης Ν. Belmont, Mass.: Athena Scientific.</span> </li> <li id="cite_note-bryc-45"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-bryc_45-0">23.0</a></sup> <sup><a href="#cite_ref-bryc_45-1">23.1</a></sup></span> <span class="reference-text">Bryc, Wlodzimierz (1995). <i>The Normal Distribution: Characterizations with Applications</i>. Springer-Verlag.</span> </li> <li id="cite_note-48"><span class="mw-cite-backlink"><a href="#cite_ref-48">↑</a></span> <span class="reference-text">Billingsley, Patrick (1999). <i>Convergence of probability measures</i> (2nd ed.). John Wiley & Sons. pp. 1–28. </span> </li> <li id="cite_note-50"><span class="mw-cite-backlink"><a href="#cite_ref-50">↑</a></span> <span class="reference-text">Dekking, Michel (2005). <i>A Modern Introduction to Probability and Statistics</i>. Springer. pp. 181–190.</span> </li> <li id="cite_note-52"><span class="mw-cite-backlink"><a href="#cite_ref-52">↑</a></span> <span class="reference-text">Yao, Kai; Gao, Jinwu (2016). "Law of Large Numbers for Uncertain Random Variables". <i>IEEE Transactions on Fuzzy Systems</i>. 24 (3): 615–621. </span> </li> <li id="cite_note-baranv2007-54"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-baranv2007_54-0">27.0</a></sup> <sup><a href="#cite_ref-baranv2007_54-1">27.1</a></sup></span> <span class="reference-text">Bárány, Imre; Vu, Van (2007). "Central limit theorems for Gaussian polytopes". <i>Annals of Probability</i>. Institute of Mathematical Statistics. 35 (4): 1593–1621.</span> </li> <li id="cite_note-dinov2008-55"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-dinov2008_55-0">28.0</a></sup> <sup><a href="#cite_ref-dinov2008_55-1">28.1</a></sup></span> <span class="reference-text">Dinov, Ivo; Christou, Nicolas; Sanchez, Juana (2008). "Central Limit Theorem: New SOCR Applet and Demonstration Activity". <i>Journal of Statistics Education</i>. ASA. 16 (2): 1–15.</span> </li> <li id="cite_note-57"><span class="mw-cite-backlink"><a href="#cite_ref-57">↑</a></span> <span class="reference-text">Broemeling, Lyle D. (1 November 2011). "An Account of Early Statistical Inference in Arab Cryptology". <i>The American Statistician</i>. 65 (4): 255-257.</span> </li> <li id="cite_note-59"><span class="mw-cite-backlink"><a href="#cite_ref-59">↑</a></span> <span class="reference-text">Debnath, L., & Basu, K. (2015). A short history of probability theory and its applications. <i>International Journal of Mathematical Education in Science and Technology</i>, 46(1), 13-39.</span> </li> <li id="cite_note-61"><span class="mw-cite-backlink"><a href="#cite_ref-61">↑</a></span> <span class="reference-text">Laplace, P. S., (1814). English edition 1951, <i>A Philosophical Essay on Probabilities</i>, New York: Dover Publications Inc.</span> </li> <li id="cite_note-64"><span class="mw-cite-backlink"><a href="#cite_ref-64">↑</a></span> <span class="reference-text">Spanos, Aris (1986). <i>Statistical foundations of econometric modelling</i>. Cambridge New York: Cambridge University Press. </span> </li> <li id="cite_note-67"><span class="mw-cite-backlink"><a href="#cite_ref-67">↑</a></span> <span class="reference-text">Cohen, L (1989). <i>An introduction to the philosophy of induction and probability</i>. Oxford New York: Clarendon Press Oxford University Press.</span> </li> <li id="cite_note-70"><span class="mw-cite-backlink"><a href="#cite_ref-70">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.nybooks.com/articles/2017/01/19/trouble-with-quantum-mechanics/">The Trouble with Quantum Mechanics</a>.</span> </li> <li id="cite_note-71"><span class="mw-cite-backlink"><a href="#cite_ref-71">↑</a></span> <span class="reference-text">Wimmel, H. (1992). <i>Quantum Physics & Observed Reality: A Critical Interpretation of Quantum Mechanics</i>. World Scientific. p. 2. <a href="/wiki/Special:%E6%9B%B8%E6%9C%AC%E4%BE%86%E6%BA%90/9789810210106" class="internal mw-magiclink-isbn">ISBN 978-981-02-1010-6</a>.</span> </li> <li id="cite_note-76"><span class="mw-cite-backlink"><a href="#cite_ref-76">↑</a></span> <span class="reference-text">Moore, David (1992). "Teaching Statistics as a Respectable Subject". In F. Gordon and S. Gordon. <i>Statistics for the Twenty-First Century</i>. Washington, DC: The Mathematical Association of America. pp. 14-25. </span> </li> <li id="cite_note-78"><span class="mw-cite-backlink"><a href="#cite_ref-78">↑</a></span> <span class="reference-text">Zar, J. H. (1984). <i>Biostatistical Analysis</i>. Prentice-Hall International, New Jersey, pp 43-45.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="拎"><span id=".E6.8B.8E"></span>拎</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%A6%82%E7%8E%87%E8%AB%96&action=edit&section=25" title="編輯小節: 拎"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r2018312">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:#f9f9f9;display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r2018311">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, 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bold;" title="連到英文網頁">（英文）</span> <a rel="nofollow" class="external text" href="https://www.springer.com/journal/10959">Journal of Theoretical Probability</a>（譯：理論概率期刊），一本講概率論<a href="/wiki/%E7%90%86%E8%AB%96" title="理論">理論</a><a href="/wiki/%E7%A0%94%E7%A9%B6" title="研究">研究</a>嘅<a href="/wiki/%E5%AD%B8%E8%A1%93%E6%9C%9F%E5%88%8A" title="學術期刊">期刊</a>。</li> <li><span style="font-family: sans-serif; cursor: default; color: var(--color-subtle, #555); font-size: 0.8em; position: relative; bottom: 0.1em; font-weight: bold;" title="連到英文網頁">（英文）</span> <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/logic-probability/">Logic and Probability</a>（譯：邏輯同概率），<a href="/wiki/%E5%8F%B2%E4%B8%B9%E7%A6%8F%E5%93%B2%E5%AD%B8%E7%99%BE%E7%A7%91%E5%85%A8%E6%9B%B8" title="史丹福哲學百科全書">史丹福哲學百科全書</a>講點樣結合邏輯同概率嘅思考。</li> <li><span style="font-family: sans-serif; cursor: default; color: var(--color-subtle, #555); font-size: 0.8em; position: relative; bottom: 0.1em; font-weight: bold;" title="連到英文網頁">（英文）</span> <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/probability-interpret/">Interpretations of Probability</a>（譯：對概率嘅詮釋），史丹福哲學百科全書講概率要點詮釋。</li></ul> <div role="navigation" class="navbox" aria-labelledby="概率論" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><div class="plainlinks hlist navbar mini"><ul><li class="nv-view"><a href="/wiki/Template:%E6%A6%82%E7%8E%87%E8%AB%96" title="Template:概率論"><abbr title="去睇呢個模" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">睇</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:%E6%A6%82%E7%8E%87%E8%AB%96&action=edit&redlink=1" class="new" title="Template talk:概率論 (無呢版)"><abbr title="傾呢個模" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">傾</abbr></a></li><li class="nv-edit"><a class="external text" href="https://zh-yue.wikipedia.org/w/index.php?title=Template:%E6%A6%82%E7%8E%87%E8%AB%96&action=edit"><abbr title="編輯呢個模" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">改</abbr></a></li></ul></div><div id="概率論" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">概率論</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="3"><div id="可以睇吓概率論詞彙表">可以睇吓<a href="/wiki/%E6%A6%82%E7%8E%87%E8%AB%96%E8%A9%9E%E5%BD%99%E8%A1%A8" class="mw-redirect" title="概率論詞彙表">概率論詞彙表</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">基礎</th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;padding-left:0em;padding-right:0em;"><div style="padding:0em 0.75em;"><a href="/wiki/%E6%A6%82%E7%8E%87" title="概率">概率</a></div></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E5%AF%A6%E9%A9%97_(%E6%A6%82%E7%8E%87%E8%AB%96)" title="實驗 (概率論)">實驗</a>（<a href="/wiki/%E4%BC%AF%E5%8A%AA%E5%88%A9%E8%A9%A6%E9%A9%97" title="伯努利試驗">伯努利</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%93%B2%E9%AA%B0%E4%BB%94" class="mw-redirect" title="擲骰仔">擲骰仔</a>同<a href="/wiki/%E6%8E%9F%E9%8A%80%E4%BB%94" title="掟銀仔">掟銀仔</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%8F%AF%E8%83%BD%E7%B5%90%E6%9E%9C" title="可能結果">可能結果</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%A8%A3%E6%9C%AC%E7%A9%BA%E9%96%93" title="樣本空間">樣本</a>同<a href="/wiki/%E6%A6%82%E7%8E%87%E7%A9%BA%E9%96%93" title="概率空間">概率</a>空間<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%AF%A6%E9%9A%9B%E6%95%B8%E5%80%BC" title="實際數值">實際數值</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;padding-left:0em;padding-right:0em;"><div style="padding:0em 0.75em;">概念</div></th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E4%B8%8D%E7%A2%BA%E5%AE%9A" title="不確定">不確定</a>同<a href="/wiki/%E9%9A%A8%E6%A9%9F" title="隨機">隨機</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E8%B3%87%E8%A8%8A" title="資訊">資訊</a>（<a href="/wiki/%E5%AE%8C%E5%85%A8%E8%B3%87%E8%A8%8A" title="完全資訊">完全資訊</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%B1%BA%E5%AE%9A%E5%9E%8B%E7%B3%BB%E7%B5%B1" title="決定型系統">決定型系統</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%A6%82%E7%8E%87%E5%85%AC%E7%90%86" title="概率公理">概率公理</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%BA%AB%E6%B0%8F%E5%9C%96" title="溫氏圖">Venn 圖</a></div></td></tr></tbody></table><div></div></td><td class="navbox-image" rowspan="8" style="width:1px;padding:0px 0px 0px 2px"><div><span typeof="mw:File"><a href="/wiki/File:Noto_Emoji_Pie_1f3b2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Noto_Emoji_Pie_1f3b2.svg/150px-Noto_Emoji_Pie_1f3b2.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Noto_Emoji_Pie_1f3b2.svg/225px-Noto_Emoji_Pie_1f3b2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/91/Noto_Emoji_Pie_1f3b2.svg/300px-Noto_Emoji_Pie_1f3b2.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E4%BA%8B%E4%BB%B6_(%E6%A6%82%E7%8E%87%E8%AB%96)" title="事件 (概率論)">事件</a></th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E5%9F%BA%E6%9C%AC%E4%BA%8B%E4%BB%B6" title="基本事件">基本事件</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%B0%8D%E7%AB%8B%E4%BA%8B%E4%BB%B6" title="對立事件">對立事件</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E4%BA%92%E8%A3%9C%E4%BA%8B%E4%BB%B6" title="互補事件">互補事件</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E4%BA%92%E6%96%A5" title="互斥">互斥</a>同<a href="/wiki/%E9%9D%9E%E4%BA%92%E6%96%A5" class="mw-redirect" title="非互斥">非互斥</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%A2%9D%E4%BB%B6%E6%A6%82%E7%8E%87" title="條件概率">條件概率</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%8D%A8%E7%AB%8B_(%E6%A6%82%E7%8E%87%E8%AB%96)" title="獨立 (概率論)">獨立</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%A2%9D%E4%BB%B6%E7%8D%A8%E7%AB%8B" title="條件獨立">條件獨立</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E9%9A%A8%E6%A9%9F%E9%81%8E%E7%A8%8B" title="隨機過程">隨機過程</a></th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E9%9A%A8%E6%A9%9F%E8%AE%8A%E6%95%B8" title="隨機變數">隨機變數</a>（<a href="/wiki/%E6%9C%9F%E6%9C%9B%E5%80%BC" title="期望值">期望值</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%9A%A8%E6%A9%9F%E8%AE%8A%E6%95%B8%E5%8C%AF%E5%90%88" title="隨機變數匯合">匯合</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%9A%A8%E6%A9%9F%E6%BC%AB%E6%AD%A5" title="隨機漫步">隨機漫步</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%A6%AC%E5%8F%AF%E5%A4%AB%E9%8F%88" title="馬可夫鏈">馬可夫鏈</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%A6%AC%E5%8F%AF%E5%A4%AB%E6%B1%BA%E7%AD%96%E9%81%8E%E7%A8%8B" title="馬可夫決策過程">馬可夫決策過程</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%B9%B3%E7%A9%A9%E9%81%8E%E7%A8%8B" title="平穩過程">平穩過程</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%A6%82%E7%8E%87%E5%88%86%E4%BD%88" title="概率分佈">概率分佈</a></th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E9%9B%A2%E6%95%A3%E6%A6%82%E7%8E%87%E5%88%86%E4%BD%88" class="mw-redirect" title="離散概率分佈">離散</a>定<a href="/wiki/%E9%80%A3%E7%BA%8C%E6%A6%82%E7%8E%87%E5%88%86%E4%BD%88" class="mw-redirect" title="連續概率分佈">連續</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E4%BC%AF%E5%8A%AA%E5%88%A9%E5%88%86%E4%BD%88" title="伯努利分佈">伯努利</a>同<a href="/wiki/%E4%BA%8C%E9%A0%85%E5%88%86%E4%BD%88" title="二項分佈">二項</a>分佈<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%B8%B8%E6%85%8B%E5%88%86%E4%BD%88" title="常態分佈">常態分佈</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E8%81%AF%E5%90%88%E6%A6%82%E7%8E%87%E5%88%86%E4%BD%88" title="聯合概率分佈">聯合概率分佈</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%A6%82%E7%8E%87%E5%88%86%E4%BD%88%E4%B8%80%E8%A6%BD" title="概率分佈一覽">概率分佈一覽</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">重要定律</th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E6%A6%82%E7%8E%87%E9%80%A3%E9%8E%96%E6%B3%95%E5%89%87" title="概率連鎖法則">概率連鎖法則</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E8%B2%9D%E8%91%89%E6%96%AF%E5%AE%9A%E7%90%86" title="貝葉斯定理">貝葉斯定理</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%A4%A7%E6%95%B8%E5%AE%9A%E5%BE%8B" title="大數定律">大數定律</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E4%B8%AD%E5%A4%AE%E6%A5%B5%E9%99%90%E5%AE%9A%E7%90%86" title="中央極限定理">中央極限定理</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%B8%83%E6%B0%8F%E4%B8%8D%E7%AD%89%E5%BC%8F" title="布氏不等式">布氏不等式</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%A6%82%E7%8E%87%E5%98%85%E8%A9%AE%E9%87%8B" title="概率嘅詮釋">詮釋</a></th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E5%8F%A4%E5%85%B8%E6%A6%82%E7%8E%87%E5%AE%9A%E7%BE%A9" title="古典概率定義">古典概率定義</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%A0%BB%E7%8E%87%E5%AD%B8%E6%B4%BE%E6%A6%82%E7%8E%87" title="頻率學派概率">頻率學派概率</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E8%B2%9D%E8%91%89%E6%96%AF%E6%A6%82%E7%8E%87" title="貝葉斯概率">貝葉斯概率</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%B1%BA%E5%AE%9A%E8%AB%96" title="決定論">決定論</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">應用</th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E8%B3%87%E8%A8%8A%E7%90%86%E8%AB%96" title="資訊理論">資訊理論</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%AF%86%E7%A2%BC%E5%AD%B8" title="密碼學">密碼學</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%B5%B1%E8%A8%88%E5%AD%B8" title="統計學">統計學</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%89%A9%E7%90%86%E5%AD%B8" title="物理學">物理學</a>（<a href="/wiki/%E9%87%8F%E5%AD%90%E5%8A%9B%E5%AD%B8" title="量子力學">量子力學</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%B5%B1%E8%A8%88%E5%8A%9B%E5%AD%B8" title="統計力學">統計力學</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%A9%9F%E6%A2%B0%E5%AD%B8%E7%BF%92" title="機械學習">機械學習</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%81%8A%E6%88%B2%E8%A8%AD%E8%A8%88" title="遊戲設計">遊戲設計</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">拉雜相關</th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E5%BF%AB%E6%80%9D%E9%82%8F%E8%BC%AF" title="快思邏輯">快思邏輯</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%A9%9F%E7%8E%87%E9%81%8A%E6%88%B2" title="機率遊戲">機率遊戲</a>同<a href="/wiki/%E8%B3%AD" title="賭">賭錢</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%9A%A8%E6%A9%9F%E6%95%B8%E7%94%9F%E6%88%90" title="隨機數生成">RNG</a> 同<a href="/wiki/%E6%93%AC%E4%BA%82%E6%95%B8%E7%94%A2%E7%94%9F" title="擬亂數產生">擬亂數產生</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E4%BC%8A%E8%97%A4%E5%BC%95%E7%90%86" title="伊藤引理">伊藤引理</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%9A%A8%E6%A9%9F%E5%BE%AE%E7%A9%8D%E5%88%86" title="隨機微積分">隨機微積分</a></div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div><a href="/wiki/Category:%E6%88%96%E7%84%B6%E7%8E%87%E8%AB%96" title="Category:或然率論">概率論類</a></div></td></tr></tbody></table></div> <div role="navigation" class="navbox" aria-labelledby="應用數學" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><div class="plainlinks hlist navbar mini"><ul><li class="nv-view"><a href="/wiki/Template:%E6%87%89%E7%94%A8%E6%95%B8%E5%AD%B8" title="Template:應用數學"><abbr title="去睇呢個模" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">睇</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:%E6%87%89%E7%94%A8%E6%95%B8%E5%AD%B8" title="Template talk:應用數學"><abbr title="傾呢個模" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">傾</abbr></a></li><li class="nv-edit"><a class="external text" href="https://zh-yue.wikipedia.org/w/index.php?title=Template:%E6%87%89%E7%94%A8%E6%95%B8%E5%AD%B8&action=edit"><abbr title="編輯呢個模" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">改</abbr></a></li></ul></div><div id="應用數學" style="font-size:114%;margin:0 4em"><a href="/wiki/%E6%87%89%E7%94%A8%E6%95%B8%E5%AD%B8" title="應用數學">應用數學</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="3"><div id="泛指數學喺科學同工程學上嘅應用">泛指<a href="/wiki/%E6%95%B8%E5%AD%B8" title="數學">數學</a>喺<a href="/wiki/%E7%A7%91%E5%AD%B8" title="科學">科學</a>同<a href="/wiki/%E5%B7%A5%E7%A8%8B%E5%AD%B8" title="工程學">工程學</a>上嘅應用</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E9%9B%A2%E6%95%A3%E6%95%B8%E5%AD%B8" title="離散數學">離散數學</a></th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E5%91%BD%E9%A1%8C%E9%82%8F%E8%BC%AF" title="命題邏輯">命題邏輯</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%9B%86%E5%90%88%E8%AB%96" title="集合論">集合論</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E4%BB%A3%E6%95%B8%E5%AD%B8" title="代數學">代數學</a>（<a href="/wiki/%E5%A0%B4_(%E6%95%B8%E5%AD%B8)" class="mw-redirect mw-disambig" title="場 (數學)">場</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%B5%84%E5%90%88%E6%95%B8%E5%AD%B8" title="組合數學">組合數學</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%9C%96%E8%AB%96" title="圖論">圖論</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%9B%A2%E6%95%A3%E5%B9%BE%E4%BD%95%E5%AD%B8" class="mw-redirect" title="離散幾何學">離散幾何學</a></div></td><td class="navbox-image" rowspan="8" style="width:1px;padding:0px 0px 0px 2px"><div><span typeof="mw:File"><a href="/wiki/File:Aizerman2d.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Aizerman2d.jpg/150px-Aizerman2d.jpg" decoding="async" width="150" height="144" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Aizerman2d.jpg/225px-Aizerman2d.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Aizerman2d.jpg/300px-Aizerman2d.jpg 2x" data-file-width="591" data-file-height="569" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%95%B8%E5%AD%B8%E5%88%86%E6%9E%90" title="數學分析">數<br />學<br />分<br />析</a></th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;padding-left:0em;padding-right:0em;"><div style="padding:0em 0.75em;">分析基礎</div></th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E5%AF%A6%E6%95%B8" title="實數">實數</a>同<a href="/wiki/%E8%99%9B%E6%95%B8" title="虛數">虛數</a>（<a href="/wiki/%E5%AF%A6%E5%88%86%E6%9E%90" title="實分析">實分析</a>同<a href="/wiki/%E8%A4%87%E5%88%86%E6%9E%90" title="複分析">複分析</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%96%93%E8%B7%9D" title="間距">間距</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%84%A1%E7%AA%AE%E7%9B%A1" title="無窮盡">無窮盡</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%84%A1%E7%AA%AE%E5%B0%8F%E9%87%8F" title="無窮小量">無窮小量</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;padding-left:0em;padding-right:0em;"><div style="padding:0em 0.75em;"><a href="/wiki/%E6%95%B8%E5%88%97" class="mw-redirect" title="數列">數列</a></div></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E5%AD%90%E6%95%B8%E5%88%97" title="子數列">子數列</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%A2%9E%E6%B8%9B%E6%95%B8%E5%88%97" title="增減數列">增減數列</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%83%9D%E6%B0%8F%E6%95%B8%E5%88%97" title="郝氏數列">郝氏數列</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E4%BF%9D%E8%A5%BF%E5%A5%B4-%E8%8F%AF%E5%AF%A6%E6%96%AF%E5%AE%9A%E7%90%86" title="保西奴-華實斯定理">保西奴-華實斯定理</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;padding-left:0em;padding-right:0em;"><div style="padding:0em 0.75em;"><a href="/wiki/%E5%87%BD%E6%95%B8" title="函數">函數</a></div></th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E9%80%A3%E7%BA%8C%E5%87%BD%E6%95%B8" title="連續函數">連續函數</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%A5%B5%E9%99%90_(%E5%87%BD%E6%95%B8)" title="極限 (函數)">極限</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%80%BC%E8%BF%91%E7%90%86%E8%AB%96" title="逼近理論">逼近理論</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%B3%9B%E5%87%BD%E5%88%86%E6%9E%90" title="泛函分析">泛函分析</a>（<a href="/wiki/%E8%AE%8A%E5%88%86%E6%B3%95" title="變分法">變分法</a>）</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;padding-left:0em;padding-right:0em;"><div style="padding:0em 0.75em;"><a href="/wiki/%E5%BE%AE%E7%A9%8D%E5%88%86" title="微積分">微積分</a></div></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="微分" scope="row" class="navbox-group" style="width:1%;padding-left:0em;padding-right:0em;"><div style="padding:0em 0.75em;"><a href="/wiki/%E5%BE%AE%E5%88%86" title="微分">微分</a></div></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="微分方程">微分方程</a>（<a href="/wiki/%E5%B8%B8%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="常微分方程">常</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%81%8F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="偏微分方程">偏</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%9A%A8%E6%A9%9F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="隨機微分方程">隨機</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E8%A4%87%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="複微分方程">複</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%BE%AE%E5%88%86%E5%B9%BE%E4%BD%95" title="微分幾何">微分幾何</a></div></td></tr><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E7%A9%8D%E5%88%86" title="積分">積分</a>（<a href="/wiki/%E9%BB%8E%E6%9B%BC%E7%A9%8D%E5%88%86" title="黎曼積分">黎曼積分</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%A9%8D%E5%88%86%E6%96%B9%E7%A8%8B" title="積分方程">積分方程</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%A4%9A%E5%85%83%E5%BE%AE%E7%A9%8D%E5%88%86" title="多元微積分">多元微積分</a>（<a href="/wiki/%E5%90%91%E9%87%8F%E5%BE%AE%E7%A9%8D%E5%88%86" title="向量微積分">向量</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%BC%B5%E9%87%8F%E5%BE%AE%E7%A9%8D%E5%88%86" title="張量微積分">張量</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%A4%9A%E9%87%8D%E7%A9%8D%E5%88%86" title="多重積分">多重積分</a>）</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;padding-left:0em;padding-right:0em;"><div style="padding:0em 0.75em;">第啲分析</div></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E5%A4%9A%E9%87%8D%E7%B7%9A%E6%80%A7%E4%BB%A3%E6%95%B8" title="多重線性代數">多線代數</a>（<a href="/wiki/%E5%90%91%E9%87%8F" title="向量">向量</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%90%91%E9%87%8F%E7%A9%BA%E9%96%93" title="向量空間">向量空間</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%BC%B5%E9%87%8F" title="張量">張量</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%8B%95%E6%85%8B%E7%B3%BB%E7%B5%B1" title="動態系統">動態系統</a>（<a href="/wiki/%E6%B7%B7%E6%B2%8C%E7%90%86%E8%AB%96" title="混沌理論">混沌理論</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%8E%A7%E5%88%B6%E7%90%86%E8%AB%96" title="控制理論">控制理論</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E8%AB%A7%E6%B3%A2%E5%88%86%E6%9E%90" title="諧波分析">諧波分析</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%95%B8%E5%80%BC%E5%88%86%E6%9E%90" title="數值分析">數值分析</a></div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E9%81%8B%E7%AE%97%E6%95%B8%E5%AD%B8" title="運算數學">運算數學</a></th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E9%81%8B%E7%AE%97%E7%90%86%E8%AB%96" title="運算理論">運算理論</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%81%8B%E7%AE%97%E9%82%8F%E8%BC%AF" title="運算邏輯">運算邏輯</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E8%87%AA%E5%8B%95%E6%A9%9F%E7%90%86%E8%AB%96" title="自動機理論">自動機理論</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%BC%94%E7%AE%97%E6%B3%95" title="演算法">演算法</a>（<a href="/wiki/%E6%BC%94%E7%AE%97%E6%B3%95%E5%88%86%E6%9E%90" title="演算法分析">分析</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E8%B3%87%E8%A8%8A%E7%90%86%E8%AB%96" title="資訊理論">資訊理論</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%AF%86%E7%A2%BC%E5%AD%B8" title="密碼學">密碼學</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%81%8B%E7%AE%97%E6%95%B8%E8%AB%96" title="運算數論">運算數論</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%A7%91%E5%AD%B8%E9%81%8B%E7%AE%97" title="科學運算">科學運算</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%9B%BB%E8%85%A6%E4%BB%A3%E6%95%B8" title="電腦代數">電腦代數</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a class="mw-selflink selflink">概率論</a></th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E9%9A%A8%E6%A9%9F%E8%AE%8A%E6%95%B8" title="隨機變數">隨機變數</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%9A%A8%E6%A9%9F%E9%81%8E%E7%A8%8B" title="隨機過程">隨機過程</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%A6%AC%E5%8F%AF%E5%A4%AB%E9%8F%88" title="馬可夫鏈">馬可夫鏈</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%A6%82%E7%8E%87%E5%88%86%E4%BD%88" title="概率分佈">概率分佈</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%9A%A8%E6%A9%9F%E5%BE%AE%E7%A9%8D%E5%88%86" title="隨機微積分">隨機微積分</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%B1%BA%E7%AD%96%E8%AB%96" title="決策論">決策論</a></th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E5%8D%9A%E5%BC%88%E8%AB%96" title="博弈論">博弈論</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%B5%B1%E8%A8%88%E5%AD%B8" title="統計學">統計學</a>（<a href="/wiki/%E6%8F%8F%E8%BF%B0%E7%B5%B1%E8%A8%88%E5%AD%B8" title="描述統計學">描述</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%8E%A8%E8%AB%96%E7%B5%B1%E8%A8%88%E5%AD%B8" title="推論統計學">推論</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%9C%80%E4%BD%B3%E5%8C%96" title="最佳化">最佳化</a>（<a href="/wiki/%E7%B7%9A%E6%80%A7%E8%A6%8F%E5%8A%83" title="線性規劃">線性</a>同<a href="/wiki/%E9%9D%9E%E7%B7%9A%E6%80%A7%E8%A6%8F%E5%8A%83" title="非線性規劃">非線性</a>規劃）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%95%B8%E7%90%86%E7%B6%93%E6%BF%9F%E5%AD%B8" title="數理經濟學">數理經濟學</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%95%B8%E7%90%86%E9%87%91%E8%9E%8D%E5%AD%B8" title="數理金融學">數理金融學</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%81%8B%E7%B1%8C%E5%AD%B8" title="運籌學">運籌學</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%B2%BE%E7%AE%97%E5%AD%B8" title="精算學">精算學</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%95%B8%E7%90%86%E7%89%A9%E7%90%86%E5%AD%B8" title="數理物理學">數理物理</a></th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E5%88%86%E6%9E%90%E5%8A%9B%E5%AD%B8" title="分析力學">分析力學</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%A0%B4_(%E7%89%A9%E7%90%86)" title="場 (物理)">場</a>（<a href="/wiki/%E8%A6%8F%E7%AF%84%E5%A0%B4%E8%AB%96" title="規範場論">規範場論</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%87%8F%E5%AD%90%E5%A0%B4%E8%AB%96" title="量子場論">量子場論</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%87%8F%E5%AD%90%E7%BE%A3" title="量子羣">量子羣</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">拉雜應用</th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E6%95%B8%E7%90%86%E5%8C%96%E5%AD%B8" title="數理化學">數理化學</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%95%B8%E7%90%86%E7%94%9F%E7%89%A9%E5%AD%B8" title="數理生物學">數理生物學</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%95%B8%E7%90%86%E5%BF%83%E7%90%86%E5%AD%B8" title="數理心理學">數理心理學</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%95%B8%E7%90%86%E7%A4%BE%E6%9C%83%E5%AD%B8" title="數理社會學">數理社會學</a>（<a href="/wiki/%E7%A4%BE%E6%9C%83%E9%81%B8%E6%93%87%E7%90%86%E8%AB%96" title="社會選擇理論">社會選擇理論</a>）</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">拉雜相關</th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E6%95%B8%E5%AD%B8%E6%A8%A1%E5%9E%8B" title="數學模型">數學模型</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%B7%A5%E7%A8%8B%E6%95%B8%E5%AD%B8" title="工程數學">工程數學</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%87%89%E7%94%A8%E7%A7%91%E5%AD%B8" title="應用科學">應用科學</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%A8%A1%E6%8E%A7%E5%AD%B8" title="模控學">模控學</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%B3%BB%E7%B5%B1" title="系統">系統</a>（<a href="/wiki/%E8%A4%87%E9%9B%9C%E7%B3%BB%E7%B5%B1" title="複雜系統">複雜系統</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%95%B8%E5%AD%B8%E5%96%BA%E8%87%AA%E7%84%B6%E7%A7%91%E5%AD%B8%E4%B8%8A%E5%98%85%E8%B6%85%E5%B8%B8%E6%95%88%E7%94%A8" title="數學喺自然科學上嘅超常效用">數學喺自然科學上嘅超常效用</a></div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div><b><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> <a href="/wiki/Portal:%E6%95%B8%E5%AD%B8" title="Portal:數學">數學主題</a></b><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/Category:%E6%87%89%E7%94%A8%E6%95%B8%E5%AD%B8" title="Category:應用數學">應用數學類</a></div></td></tr></tbody></table></div> <div role="navigation" class="navbox" aria-labelledby="現代數學同數學各領域" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" 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padding:0;">改</abbr></a></li></ul></div><div id="現代數學同數學各領域" style="font-size:114%;margin:0 4em">現代<a href="/wiki/%E6%95%B8%E5%AD%B8" title="數學">數學</a>同數學各領域</div></th></tr><tr><td class="navbox-abovebelow" colspan="3"><div id="純粹同應用數學_·_數學史_·_數學之美"><a href="/wiki/%E7%B4%94%E7%B2%B9%E6%95%B8%E5%AD%B8" title="純粹數學">純粹</a>同<a href="/wiki/%E6%87%89%E7%94%A8%E6%95%B8%E5%AD%B8" title="應用數學">應用</a>數學<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%95%B8%E5%AD%B8%E5%8F%B2" title="數學史">數學史</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%95%B8%E5%AD%B8%E4%B9%8B%E7%BE%8E" title="數學之美">數學之美</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E" title="數學證明">數學證明</a></th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%9B%B4%E6%8E%A5%E8%AD%89%E6%98%8E" title="直接證明">直接證明</a></li> <li><a href="/wiki/%E6%95%B8%E5%AD%B8%E6%AD%B8%E7%B4%8D%E6%B3%95" title="數學歸納法">數學歸納法</a></li> <li><a href="/wiki/%E5%90%A6%E5%AE%9A%E8%AD%89%E6%98%8E" title="否定證明">否定證明</a></li> <li><a href="/wiki/%E5%8F%8D%E8%AD%89%E6%B3%95" title="反證法">反證法</a></li> <li><a href="/wiki/%E6%A7%8B%E9%80%A0%E6%B3%95" title="構造法">構造法</a></li> <li><a href="/wiki/%E5%88%86%E9%A1%9E%E8%AD%89%E6%98%8E" title="分類證明">分類證明</a></li></ul> </div></td><td class="navbox-image" rowspan="10" style="width:1px;padding:0px 0px 0px 2px"><div><span typeof="mw:File"><a href="/wiki/File:Mathematics.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Mathematics.png/150px-Mathematics.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Mathematics.png/225px-Mathematics.png 1.5x, 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title="數學符號">數學符號</a></li> <li><a href="/wiki/%E6%95%B8%E5%AD%B8%E7%89%A9%E9%AB%94" title="數學物體">數學物體</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%95%B8%E8%AB%96" title="數論">數論</a></th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E6%95%B8" title="數">數</a>（<a href="/wiki/%E8%B3%AA%E6%95%B8" title="質數">質數</a>）</li> <li><a href="/wiki/%E7%AE%97%E8%A1%93" title="算術">算術</a></li> <li><a href="/wiki/%E4%BB%A3%E6%95%B8%E6%95%B8%E8%AB%96" title="代數數論">代數數論</a></li> <li><a href="/wiki/%E8%A7%A3%E6%9E%90%E6%95%B8%E8%AB%96" title="解析數論">解析數論</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E4%BB%A3%E6%95%B8%E5%AD%B8" title="代數學">代數學</a></th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%9F%BA%E6%9C%AC%E4%BB%A3%E6%95%B8" title="基本代數">基本代數</a></li> <li><a href="/wiki/%E7%B7%9A%E6%80%A7%E4%BB%A3%E6%95%B8" title="線性代數">線性代數</a>（<a href="/wiki/%E5%A4%9A%E9%87%8D%E7%B7%9A%E6%80%A7%E4%BB%A3%E6%95%B8" title="多重線性代數">多重</a>）</li> <li><a href="/wiki/%E6%8A%BD%E8%B1%A1%E4%BB%A3%E6%95%B8" title="抽象代數">抽象代數</a>（<a href="/wiki/%E4%BA%A4%E6%8F%9B%E4%BB%A3%E6%95%B8" title="交換代數">交換代數</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%BE%A3%E8%AB%96" title="羣論">羣論</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E4%BC%BD%E8%8F%AF%E7%90%86%E8%AB%96" title="伽華理論">伽華理論</a>）</li> <li><a href="/wiki/%E5%A0%B4%E8%AB%96" title="場論">場論</a></li> <li><a href="/wiki/%E6%B3%9B%E4%BB%A3%E6%95%B8" title="泛代數">泛代數</a></li> <li><a href="/wiki/%E5%90%8C%E8%AA%BF%E4%BB%A3%E6%95%B8" 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