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data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 42 languages" > <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-42" 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">42 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AD%D8%AF%D8%AB_(%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%D8%A7%D8%AA)" title="حدث (نظرية الاحتمالات) – Arabic" lang="ar" hreflang="ar" data-title="حدث (نظرية الاحتمالات)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/T%C9%99sad%C3%BCfi_hadis%C9%99" title="Təsadüfi hadisə – Azerbaijani" lang="az" hreflang="az" data-title="Təsadüfi hadisə" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%98%E0%A6%9F%E0%A6%A8%E0%A6%BE_(%E0%A6%B8%E0%A6%AE%E0%A7%8D%E0%A6%AD%E0%A6%BE%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="ঘটনা (সম্ভাবনা তত্ত্ব) – Bangla" lang="bn" hreflang="bn" data-title="ঘটনা (সম্ভাবনা তত্ত্ব)" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%92%D1%8B%D0%BF%D0%B0%D0%B4%D0%BA%D0%BE%D0%B2%D0%B0%D1%8F_%D0%BF%D0%B0%D0%B4%D0%B7%D0%B5%D1%8F" title="Выпадковая падзея – Belarusian" lang="be" hreflang="be" data-title="Выпадковая падзея" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Succ%C3%A9s" title="Succés – Catalan" lang="ca" hreflang="ca" data-title="Succés" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%C4%82%D0%BD%D1%81%C4%83%D1%80%D1%82_%D0%BF%D1%83%D0%BB%C4%83%D0%BC" title="Ăнсăрт пулăм – Chuvash" lang="cv" hreflang="cv" data-title="Ăнсăрт пулăм" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/N%C3%A1hodn%C3%BD_jev" title="Náhodný jev – Czech" lang="cs" hreflang="cs" data-title="Náhodný jev" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Digwyddiad_(theori_tebygolrwydd)" title="Digwyddiad (theori tebygolrwydd) – Welsh" lang="cy" hreflang="cy" data-title="Digwyddiad (theori tebygolrwydd)" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Ereignis_(Wahrscheinlichkeitstheorie)" title="Ereignis (Wahrscheinlichkeitstheorie) – German" lang="de" hreflang="de" data-title="Ereignis (Wahrscheinlichkeitstheorie)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/S%C3%BCndmus_(matemaatika)" title="Sündmus (matemaatika) – Estonian" lang="et" hreflang="et" data-title="Sündmus (matemaatika)" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%93%CE%B5%CE%B3%CE%BF%CE%BD%CF%8C%CF%82_(%CE%B8%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="Γεγονός (θεωρία πιθανοτήτων) – Greek" lang="el" hreflang="el" data-title="Γεγονός (θεωρία πιθανοτήτων)" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Evento_aleatorio" title="Evento aleatorio – Spanish" lang="es" hreflang="es" data-title="Evento aleatorio" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Evento_(probablo-teorio)" title="Evento (probablo-teorio) – Esperanto" lang="eo" hreflang="eo" data-title="Evento (probablo-teorio)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Gertakizun_(probabilitatea)" title="Gertakizun (probabilitatea) – Basque" lang="eu" hreflang="eu" data-title="Gertakizun (probabilitatea)" data-language-autonym="Euskara" data-language-local-name="Basque" 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%BE%DB%8C%D8%B4%D8%A7%D9%85%D8%AF" title="پیشامد – Persian" lang="fa" hreflang="fa" data-title="پیشامد" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/%C3%89v%C3%A9nement_(probabilit%C3%A9s)" title="Événement (probabilités) – French" lang="fr" hreflang="fr" data-title="Événement (probabilités)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%82%AC%EA%B1%B4_(%ED%99%95%EB%A5%A0%EB%A1%A0)" title="사건 (확률론) – Korean" lang="ko" hreflang="ko" data-title="사건 (확률론)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Atbur%C3%B0ur_(l%C3%ADkindafr%C3%A6%C3%B0i)" title="Atburður (líkindafræði) – Icelandic" lang="is" hreflang="is" data-title="Atburður (líkindafræði)" data-language-autonym="Íslenska" data-language-local-name="Icelandic" 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/Evento_(teoria_della_probabilit%C3%A0)" title="Evento (teoria della probabilità) – Italian" lang="it" hreflang="it" data-title="Evento (teoria della probabilità)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%90%D7%95%D7%A8%D7%A2" title="מאורע – Hebrew" lang="he" hreflang="he" data-title="מאורע" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%AE%E1%83%93%E1%83%9D%E1%83%9B%E1%83%98%E1%83%9A%E1%83%9D%E1%83%91%E1%83%90" title="ხდომილობა – Georgian" lang="ka" hreflang="ka" data-title="ხდომილობა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Esem%C3%A9ny_(matematika)" title="Esemény (matematika) – Hungarian" lang="hu" hreflang="hu" data-title="Esemény (matematika)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Gebeurtenis_(kansrekening)" title="Gebeurtenis (kansrekening) – Dutch" lang="nl" hreflang="nl" data-title="Gebeurtenis (kansrekening)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%BA%8B%E8%B1%A1_(%E7%A2%BA%E7%8E%87%E8%AB%96)" title="事象 (確率論) – Japanese" lang="ja" hreflang="ja" data-title="事象 (確率論)" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Hendelse" title="Hendelse – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Hendelse" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Zdarzenie_losowe_(teoria_prawdopodobie%C5%84stwa)" title="Zdarzenie losowe (teoria prawdopodobieństwa) – Polish" lang="pl" hreflang="pl" data-title="Zdarzenie losowe (teoria prawdopodobieństwa)" data-language-autonym="Polski" data-language-local-name="Polish" 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/Evento_(teoria_das_probabilidades)" title="Evento (teoria das probabilidades) – Portuguese" lang="pt" hreflang="pt" data-title="Evento (teoria das probabilidades)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%BE%D0%B5_%D1%81%D0%BE%D0%B1%D1%8B%D1%82%D0%B8%D0%B5" title="Случайное событие – Russian" lang="ru" hreflang="ru" data-title="Случайное событие" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Dogodek_(verjetnostni_ra%C4%8Dun)" title="Dogodek (verjetnostni račun) – Slovenian" lang="sl" hreflang="sl" data-title="Dogodek (verjetnostni račun)" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%95%D9%88%D9%88%D8%AF%D8%A7%D9%88_(%D8%AA%DB%8C%DB%86%D8%B1%DB%8C%DB%8C_%D8%A6%DB%95%DA%AF%DB%95%D8%B1)" title="ڕووداو (تیۆریی ئەگەر) – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ڕووداو (تیۆریی ئەگەر)" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%94%D0%BE%D0%B3%D0%B0%D1%92%D0%B0%D1%98_(%D1%82%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="Догађај (теорија вероватноће) – Serbian" lang="sr" hreflang="sr" data-title="Догађај (теорија вероватноће)" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Tapahtuma_(todenn%C3%A4k%C3%B6isyys)" title="Tapahtuma (todennäköisyys) – Finnish" lang="fi" hreflang="fi" data-title="Tapahtuma (todennäköisyys)" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/H%C3%A4ndelse_(utfall)" title="Händelse (utfall) – Swedish" lang="sv" hreflang="sv" data-title="Händelse (utfall)" data-language-autonym="Svenska" data-language-local-name="Swedish" 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%9A%E0%AF%8D%E0%AE%9A%E0%AE%BF_(%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)" title="நிகழ்ச்சி (நிகழ்தகவு) – Tamil" lang="ta" hreflang="ta" data-title="நிகழ்ச்சி (நிகழ்தகவு)" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%AB%E0%B8%95%E0%B8%B8%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%93%E0%B9%8C_(%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="เหตุการณ์ (ความน่าจะเป็น) – Thai" lang="th" hreflang="th" data-title="เหตุการณ์ (ความน่าจะเป็น)" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Olay" title="Olay – Turkish" lang="tr" hreflang="tr" data-title="Olay" data-language-autonym="Türkçe" data-language-local-name="Turkish" 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%92%D0%B8%D0%BF%D0%B0%D0%B4%D0%BA%D0%BE%D0%B2%D0%B0_%D0%BF%D0%BE%D0%B4%D1%96%D1%8F" title="Випадкова подія – Ukrainian" lang="uk" hreflang="uk" data-title="Випадкова подія" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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%88%D8%A7%D9%82%D8%B9%DB%81_(%D8%A7%D8%AD%D8%AA%D9%85%D8%A7%D9%84_%D9%86%D8%B8%D8%B1%DB%8C%DB%81)" title="واقعہ (احتمال نظریہ) – Urdu" lang="ur" hreflang="ur" data-title="واقعہ (احتمال نظریہ)" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Bi%E1%BA%BFn_c%E1%BB%91_(l%C3%BD_thuy%E1%BA%BFt_x%C3%A1c_su%E1%BA%A5t)" title="Biến cố (lý thuyết xác suất) – Vietnamese" lang="vi" hreflang="vi" data-title="Biến cố (lý thuyết xác suất)" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" 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/%E4%BA%8B%E4%BB%B6%EF%BC%88%E6%A6%82%E7%8E%87%E8%AE%BA%EF%BC%89" title="事件（概率论） – Wu" lang="wuu" hreflang="wuu" data-title="事件（概率论）" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E4%BA%8B%E4%BB%B6_(%E6%A6%82%E7%8E%87%E8%AB%96)" title="事件 (概率論) – Cantonese" lang="yue" hreflang="yue" data-title="事件 (概率論)" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%BA%8B%E4%BB%B6_(%E6%A6%82%E7%8E%87%E8%AE%BA)" title="事件 (概率论) – Chinese" lang="zh" hreflang="zh" data-title="事件 (概率论)" data-language-autonym="中文" data-language-local-name="Chinese" 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.sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on <a href="/wiki/Statistics" title="Statistics">statistics</a></td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a></th></tr><tr><td class="sidebar-image"><span typeof="mw:File"><a href="/wiki/File:Standard_deviation_diagram_micro.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Standard_deviation_diagram_micro.svg/250px-Standard_deviation_diagram_micro.svg.png" decoding="async" width="250" height="125" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Standard_deviation_diagram_micro.svg/375px-Standard_deviation_diagram_micro.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Standard_deviation_diagram_micro.svg/500px-Standard_deviation_diagram_micro.svg.png 2x" data-file-width="400" data-file-height="200" /></a></span></td></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Probability" title="Probability">Probability</a> <ul><li><a href="/wiki/Probability_axioms" title="Probability axioms">Axioms</a></li></ul></li> <li><a href="/wiki/Determinism" title="Determinism">Determinism</a> <ul><li><a href="/wiki/Deterministic_system" title="Deterministic system">System</a></li></ul></li> <li><a href="/wiki/Indeterminism" title="Indeterminism">Indeterminism</a></li> <li><a href="/wiki/Randomness" title="Randomness">Randomness</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Probability_space" title="Probability space">Probability space</a></li> <li><a href="/wiki/Sample_space" title="Sample space">Sample space</a></li> <li><a class="mw-selflink selflink">Event</a> <ul><li><a href="/wiki/Collectively_exhaustive_events" title="Collectively exhaustive events">Collectively exhaustive events</a></li> <li><a href="/wiki/Elementary_event" title="Elementary event">Elementary event</a></li> <li><a href="/wiki/Mutual_exclusivity" title="Mutual exclusivity">Mutual exclusivity</a></li> <li><a href="/wiki/Outcome_(probability)" title="Outcome (probability)">Outcome</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li></ul></li> <li><a href="/wiki/Experiment_(probability_theory)" title="Experiment (probability theory)">Experiment</a> <ul><li><a href="/wiki/Bernoulli_trial" title="Bernoulli trial">Bernoulli trial</a></li></ul></li> <li><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distribution</a> <ul><li><a href="/wiki/Bernoulli_distribution" title="Bernoulli distribution">Bernoulli distribution</a></li> <li><a href="/wiki/Binomial_distribution" title="Binomial distribution">Binomial distribution</a></li> <li><a href="/wiki/Exponential_distribution" title="Exponential distribution">Exponential distribution</a></li> <li><a href="/wiki/Normal_distribution" title="Normal distribution">Normal distribution</a></li> <li><a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto distribution</a></li> <li><a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson distribution</a></li></ul></li> <li><a href="/wiki/Probability_measure" title="Probability measure">Probability measure</a></li> <li><a href="/wiki/Random_variable" title="Random variable">Random variable</a> <ul><li><a href="/wiki/Bernoulli_process" title="Bernoulli process">Bernoulli process</a></li> <li><a href="/wiki/Continuous_or_discrete_variable" title="Continuous or discrete variable">Continuous or discrete</a></li> <li><a href="/wiki/Expected_value" title="Expected value">Expected value</a></li> <li><a href="/wiki/Variance" title="Variance">Variance</a></li> <li><a href="/wiki/Markov_chain" title="Markov chain">Markov chain</a></li> <li><a href="/wiki/Realization_(probability)" title="Realization (probability)">Observed value</a></li> <li><a href="/wiki/Random_walk" title="Random walk">Random walk</a></li> <li><a href="/wiki/Stochastic_process" title="Stochastic process">Stochastic process</a></li></ul></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Complementary_event" title="Complementary event">Complementary event</a></li> <li><a href="/wiki/Joint_probability_distribution" title="Joint probability distribution">Joint probability</a></li> <li><a href="/wiki/Marginal_distribution" title="Marginal distribution">Marginal probability</a></li> <li><a href="/wiki/Conditional_probability" title="Conditional probability">Conditional probability</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Independence_(probability_theory)" title="Independence (probability theory)">Independence</a></li> <li><a href="/wiki/Conditional_independence" title="Conditional independence">Conditional independence</a></li> <li><a href="/wiki/Law_of_total_probability" title="Law of total probability">Law of total probability</a></li> <li><a href="/wiki/Law_of_large_numbers" title="Law of large numbers">Law of large numbers</a></li> <li><a href="/wiki/Bayes%27_theorem" title="Bayes' theorem">Bayes' theorem</a></li> <li><a href="/wiki/Boole%27s_inequality" title="Boole's inequality">Boole's inequality</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li> <li><a href="/wiki/Tree_diagram_(probability_theory)" title="Tree diagram (probability theory)">Tree diagram</a></li></ul></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Probability_fundamentals" title="Template:Probability fundamentals"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Probability_fundamentals" title="Template talk:Probability fundamentals"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Probability_fundamentals" title="Special:EditPage/Template:Probability fundamentals"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, an <b>event</b> is a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of <a href="/wiki/Outcome_(probability)" title="Outcome (probability)">outcomes</a> of an <a href="/wiki/Experiment_(probability_theory)" title="Experiment (probability theory)">experiment</a> (a <a href="/wiki/Subset" title="Subset">subset</a> of the <a href="/wiki/Sample_space" title="Sample space">sample space</a>) to which a probability is assigned.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> A single outcome may be an element of many different events,<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> and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> An event consisting of only a single outcome is called an <em><a href="/wiki/Elementary_event" title="Elementary event">elementary event</a></em> or an <em>atomic event</em>; that is, it is a <a href="/wiki/Singleton_set" class="mw-redirect" title="Singleton set">singleton set</a>. An event that has more than one possible outcome is called a <b>compound event.</b> An event <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> is said to <em>occur</em> if <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> contains the outcome <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> of the <a href="/wiki/Experiment_(probability_theory)" title="Experiment (probability theory)">experiment</a> (or trial) (that is, if <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\in S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51186ba8afb2067573a9082d55dd383df1ea9214" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.67ex; height:2.176ex;" alt="{\displaystyle x\in S}"></span>).<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The probability (with respect to some <a href="/wiki/Probability_measure" title="Probability measure">probability measure</a>) that an event <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> occurs is the probability that <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> contains the outcome <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> of an experiment (that is, it is the probability that <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\in S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51186ba8afb2067573a9082d55dd383df1ea9214" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.67ex; height:2.176ex;" alt="{\displaystyle x\in S}"></span>). An event defines a <a href="/wiki/Complementary_event" title="Complementary event">complementary event</a>, namely the complementary set (the event <em>not</em> occurring), and together these define a <a href="/wiki/Bernoulli_trial" title="Bernoulli trial">Bernoulli trial</a>: did the event occur or not? </p><p>Typically, when the <a href="/wiki/Sample_space" title="Sample space">sample space</a> is finite, any subset of the sample space is an event (that is, all elements of the <a href="/wiki/Power_set" title="Power set">power set</a> of the sample space are defined as events).<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> However, this approach does not work well in cases where the sample space is <a href="/wiki/Uncountably_infinite" class="mw-redirect" title="Uncountably infinite">uncountably infinite</a>. So, when defining a <a href="/wiki/Probability_space" title="Probability space">probability space</a> it is possible, and often necessary, to exclude certain subsets of the sample space from being events (see <a href="#Events_in_probability_spaces">§ Events in probability spaces</a>, below). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="A_simple_example">A simple example</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Event_(probability_theory)&action=edit&section=1" title="Edit section: A simple example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If we assemble a deck of 52 <a href="/wiki/Playing_card" title="Playing card">playing cards</a> with no jokers, and draw a single card from the deck, then the sample space is a 52-element set, as each card is a possible outcome. An event, however, is any subset of the sample space, including any <a href="/wiki/Singleton_set" class="mw-redirect" title="Singleton set">singleton set</a> (an <a href="/wiki/Elementary_event" title="Elementary event">elementary event</a>), the <a href="/wiki/Empty_set" title="Empty set">empty set</a> (an impossible event, with probability zero) and the sample space itself (a certain event, with probability one). Other events are <a href="/wiki/Proper_subset" class="mw-redirect" title="Proper subset">proper subsets</a> of the sample space that contain multiple elements. So, for example, potential events include: </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Venn_A_subset_B.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Venn_A_subset_B.svg/150px-Venn_A_subset_B.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Venn_A_subset_B.svg/225px-Venn_A_subset_B.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Venn_A_subset_B.svg/300px-Venn_A_subset_B.svg.png 2x" data-file-width="155" data-file-height="155" /></a><figcaption>An <a href="/wiki/Euler_diagram" title="Euler diagram">Euler diagram</a> of an event. <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> is the sample space and <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> is an event.<br />By the ratio of their areas, the probability of <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> is approximately 0.4.</figcaption></figure> <ul><li>"Red and black at the same time without being a joker" (0 elements),</li> <li>"The 5 of Hearts" (1 element),</li> <li>"A King" (4 elements),</li> <li>"A Face card" (12 elements),</li> <li>"A Spade" (13 elements),</li> <li>"A Face card or a red suit" (32 elements),</li> <li>"A card" (52 elements).</li></ul> <p>Since all events are sets, they are usually written as sets (for example, {1, 2, 3}), and represented graphically using <a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagrams</a>. In the situation where each outcome in the sample space Ω is equally likely, the probability <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> of an event <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> is the following <style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="formula"></span><span class="vanchor-text">formula</span></span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {P} (A)={\frac {|A|}{|\Omega |}}\,\ \left({\text{alternatively:}}\ \Pr(A)={\frac {|A|}{|\Omega |}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mtext> </mtext> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>alternatively:</mtext> </mrow> <mtext> </mtext> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {P} (A)={\frac {|A|}{|\Omega |}}\,\ \left({\text{alternatively:}}\ \Pr(A)={\frac {|A|}{|\Omega |}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec74ab5f26af2ae71fafce7700dd81be3946dd8b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:44.314ex; height:6.509ex;" alt="{\displaystyle \mathrm {P} (A)={\frac {|A|}{|\Omega |}}\,\ \left({\text{alternatively:}}\ \Pr(A)={\frac {|A|}{|\Omega |}}\right)}"></span> This rule can readily be applied to each of the example events above. </p> <div class="mw-heading mw-heading2"><h2 id="Events_in_probability_spaces">Events in probability spaces</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Event_(probability_theory)&action=edit&section=2" title="Edit section: Events in probability spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Defining all subsets of the sample space as events works well when there are only finitely many outcomes, but gives rise to problems when the sample space is infinite. For many standard <a href="/wiki/Probability_distributions" class="mw-redirect" title="Probability distributions">probability distributions</a>, such as the <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a>, the sample space is the set of real numbers or some subset of the <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a>. Attempts to define probabilities for all subsets of the real numbers run into difficulties when one considers <a href="/wiki/Pathological_(mathematics)" title="Pathological (mathematics)">'badly behaved'</a> sets, such as those that are <a href="/wiki/Nonmeasurable_set" class="mw-redirect" title="Nonmeasurable set">nonmeasurable</a>. Hence, it is necessary to restrict attention to a more limited family of subsets. For the standard tools of probability theory, such as <a href="/wiki/Joint_probability" class="mw-redirect" title="Joint probability">joint</a> and <a href="/wiki/Conditional_probability" title="Conditional probability">conditional probabilities</a>, to work, it is necessary to use a <a href="/wiki/Sigma-algebra" class="mw-redirect" title="Sigma-algebra">σ-algebra</a>, that is, a family closed under complementation and countable unions of its members. The most natural choice of <a href="/wiki/Sigma-algebra" class="mw-redirect" title="Sigma-algebra">σ-algebra</a> is the <a href="/wiki/Borel_measure" title="Borel measure">Borel measurable</a> set derived from unions and intersections of intervals. However, the larger class of <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue measurable</a> sets proves more useful in practice. </p><p>In the general <a href="/wiki/Measure_theory" class="mw-redirect" title="Measure theory">measure-theoretic</a> description of <a href="/wiki/Probability_space" title="Probability space">probability spaces</a>, an event may be defined as an element of a selected <a href="/wiki/Sigma-algebra" class="mw-redirect" title="Sigma-algebra">𝜎-algebra</a> of subsets of the sample space. Under this definition, any subset of the sample space that is not an element of the 𝜎-algebra is not an event, and does not have a probability. With a reasonable specification of the probability space, however, all <em>events of interest</em> are elements of the 𝜎-algebra. </p> <div class="mw-heading mw-heading2"><h2 id="A_note_on_notation">A note on notation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Event_(probability_theory)&action=edit&section=3" title="Edit section: A note on notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Even though events are subsets of some sample space <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 \Omega ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9e6a34b6c6b77b92a65f692aad0f65b10f5bf26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \Omega ,}"></span> they are often written as predicates or indicators involving <a href="/wiki/Random_variable" title="Random variable">random variables</a>. For example, if <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/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> is a real-valued random variable defined on the sample space <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 \Omega ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9e6a34b6c6b77b92a65f692aad0f65b10f5bf26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \Omega ,}"></span> the event <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\omega \in \Omega \mid u<X(\omega )\leq v\}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>ω</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>∣</mo> <mi>u</mi> <mo><</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>v</mi> <mo fence="false" stretchy="false">}</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\omega \in \Omega \mid u<X(\omega )\leq v\}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c106c4afd2824083f1f01ca7c617bd17e7696c8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.503ex; height:2.843ex;" alt="{\displaystyle \{\omega \in \Omega \mid u<X(\omega )\leq v\}\,}"></span> can be written more conveniently as, simply, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u<X\leq v\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo><</mo> <mi>X</mi> <mo>≤</mo> <mi>v</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u<X\leq v\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40b23f6e369f59e76d9f4ca5075562a116324425" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.668ex; height:2.343ex;" alt="{\displaystyle u<X\leq v\,.}"></span> This is especially common in formulas for a <a href="/wiki/Probability" title="Probability">probability</a>, such as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(u<X\leq v)=F(v)-F(u)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo><</mo> <mi>X</mi> <mo>≤</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(u<X\leq v)=F(v)-F(u)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88e14ed850dcd3d661762b46681f39fa9ecc00ca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.468ex; height:2.843ex;" alt="{\displaystyle \Pr(u<X\leq v)=F(v)-F(u)\,.}"></span> The <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</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 u<X\leq v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo><</mo> <mi>X</mi> <mo>≤</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u<X\leq v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d37b121a025ad9381f693d41bf9af852a3c288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.634ex; height:2.343ex;" alt="{\displaystyle u<X\leq v}"></span> is an example of an <a href="/wiki/Inverse_image" class="mw-redirect" title="Inverse image">inverse image</a> under the <a href="/wiki/Map_(mathematics)" title="Map (mathematics)">mapping</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 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/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> because <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 \omega \in X^{-1}((u,v])}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>∈</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega \in X^{-1}((u,v])}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df27f32adf9e5444ae9926d66af8bd2fc6151c58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.468ex; height:3.176ex;" alt="{\displaystyle \omega \in X^{-1}((u,v])}"></span> if and only if <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<X(\omega )\leq v.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo><</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>v</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u<X(\omega )\leq v.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca59bce9621c6bf9a13ec8e1252056897699544d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.536ex; height:2.843ex;" alt="{\displaystyle u<X(\omega )\leq v.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Event_(probability_theory)&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Atom_(measure_theory)" title="Atom (measure theory)">Atom (measure theory)</a> – A measurable set with positive measure that contains no subset of smaller positive measure</li> <li><a href="/wiki/Complementary_event" title="Complementary event">Complementary event</a> – Opposite of a probability event</li> <li><a href="/wiki/Elementary_event" title="Elementary event">Elementary event</a> – An event that contains only one outcome<span style="display:none" class="category-wikidata-fallback-annotation">Pages displaying wikidata descriptions as a fallback</span></li> <li><a href="/wiki/Independent_event" class="mw-redirect" title="Independent event">Independent event</a> – When the occurrence of one event does not affect the likelihood of another<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Outcome_(probability)" title="Outcome (probability)">Outcome (probability)</a> – Possible result of an experiment or trial</li> <li><a href="/wiki/Pairwise_independence" title="Pairwise independence">Pairwise independent events</a> – Set of random variables of which any two are independent</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Event_(probability_theory)&action=edit&section=5" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.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"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFLeon-Garcia2008" class="citation book cs1">Leon-Garcia, Alberto (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GUJosCkbBywC"><i>Probability, statistics and random processes for electrical engineering</i></a>. Upper Saddle River, NJ: Pearson. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780131471221" title="Special:BookSources/9780131471221"><bdi>9780131471221</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability%2C+statistics+and+random+processes+for+electrical+engineering&rft.place=Upper+Saddle+River%2C+NJ&rft.pub=Pearson&rft.date=2008&rft.isbn=9780131471221&rft.aulast=Leon-Garcia&rft.aufirst=Alberto&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DGUJosCkbBywC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEvent+%28probability+theory%29" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPfeiffer1978" class="citation book cs1">Pfeiffer, Paul E. (1978). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_mayRBczVRwC&pg=PA18"><i>Concepts of probability theory</i></a>. Dover Publications. p. 18. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-63677-1" title="Special:BookSources/978-0-486-63677-1"><bdi>978-0-486-63677-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Concepts+of+probability+theory&rft.pages=18&rft.pub=Dover+Publications&rft.date=1978&rft.isbn=978-0-486-63677-1&rft.aulast=Pfeiffer&rft.aufirst=Paul+E.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D_mayRBczVRwC%26pg%3DPA18&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEvent+%28probability+theory%29" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFoerster2006" class="citation book cs1">Foerster, Paul A. (2006). <a rel="nofollow" class="external text" href="https://archive.org/details/algebratrigonome00paul_0/page/634"><i>Algebra and trigonometry: Functions and Applications, Teacher's edition</i></a> (Classics ed.). Upper Saddle River, NJ: <a href="/wiki/Prentice_Hall" title="Prentice Hall">Prentice Hall</a>. p. <a rel="nofollow" class="external text" href="https://archive.org/details/algebratrigonome00paul_0/page/634">634</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-165711-9" title="Special:BookSources/0-13-165711-9"><bdi>0-13-165711-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra+and+trigonometry%3A+Functions+and+Applications%2C+Teacher%27s+edition&rft.place=Upper+Saddle+River%2C+NJ&rft.pages=634&rft.edition=Classics&rft.pub=Prentice+Hall&rft.date=2006&rft.isbn=0-13-165711-9&rft.aulast=Foerster&rft.aufirst=Paul+A.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Falgebratrigonome00paul_0%2Fpage%2F634&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEvent+%28probability+theory%29" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDekkingKraaikampLopuhaäLudolf_Erwin2005" class="citation book cs1">Dekking, Frederik Michel; Kraaikamp, Cornelis; Lopuhaä, Hendrik Paul; Ludolf Erwin, Meester (2005). Dekking, Michel (ed.). <a rel="nofollow" class="external text" href="https://link.springer.com/book/10.1007/1-84628-168-7"><i>A modern introduction to probability and statistics: understanding why and how</i></a>. Springer texts in statistics. London [Heidelberg]: Springer. p. 14. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-85233-896-1" title="Special:BookSources/978-1-85233-896-1"><bdi>978-1-85233-896-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+modern+introduction+to+probability+and+statistics%3A+understanding+why+and+how&rft.place=London+%5BHeidelberg%5D&rft.series=Springer+texts+in+statistics&rft.pages=14&rft.pub=Springer&rft.date=2005&rft.isbn=978-1-85233-896-1&rft.aulast=Dekking&rft.aufirst=Frederik+Michel&rft.au=Kraaikamp%2C+Cornelis&rft.au=Lopuha%C3%A4%2C+Hendrik+Paul&rft.au=Ludolf+Erwin%2C+Meester&rft_id=https%3A%2F%2Flink.springer.com%2Fbook%2F10.1007%2F1-84628-168-7&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEvent+%28probability+theory%29" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFŠirjaev2016" class="citation book cs1">Širjaev, Alʹbert N. (2016). <i>Probability-1</i>. Graduate texts in mathematics. Translated by Boas, Ralph Philip; Chibisov, Dmitry (3rd ed.). New York Heidelberg Dordrecht London: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-72205-4" title="Special:BookSources/978-0-387-72205-4"><bdi>978-0-387-72205-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability-1&rft.place=New+York+Heidelberg+Dordrecht+London&rft.series=Graduate+texts+in+mathematics&rft.edition=3rd&rft.pub=Springer&rft.date=2016&rft.isbn=978-0-387-72205-4&rft.aulast=%C5%A0irjaev&rft.aufirst=Al%CA%B9bert+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEvent+%28probability+theory%29" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Event_(probability_theory)&action=edit&section=6" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.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:var(--background-color-interactive-subtle,#f8f9fa);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;min-width:0}}@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><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.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/en/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/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; 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