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vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Vaihda alaosio Joukko-opillinen määritelmä</span> </button> <ul id="toc-Joukko-opillinen_määritelmä-sublist" class="vector-toc-list"> <li id="toc-Esimerkkejä" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Esimerkkejä"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Esimerkkejä</span> </div> </a> <ul id="toc-Esimerkkejä-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Komplementtisääntö" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Komplementtisääntö"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Komplementtisääntö</span> </div> </a> <ul id="toc-Komplementtisääntö-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lähteet" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lähteet"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Lähteet</span> </div> </a> <ul id="toc-Lähteet-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="Sisällysluettelo" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Sisällysluettelo" > <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="Vaihda sisällysluettelo" > <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">Vaihda sisällysluettelo</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">Vastatapahtuma</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="Mene artikkeliin toisella kielellä. Saatavilla 10 kielellä" > <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-10" 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">10 kieltä</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D1%82%D0%B8%D0%B2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B6%D0%BD%D0%BE_%D1%81%D1%8A%D0%B1%D0%B8%D1%82%D0%B8%D0%B5" title="Противоположно събитие — bulgaria" lang="bg" hreflang="bg" data-title="Противоположно събитие" data-language-autonym="Български" data-language-local-name="bulgaria" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Esdeveniment_contrari" title="Esdeveniment contrari — katalaani" lang="ca" hreflang="ca" data-title="Esdeveniment contrari" data-language-autonym="Català" data-language-local-name="katalaani" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Complementary_event" title="Complementary event — englanti" lang="en" hreflang="en" data-title="Complementary event" data-language-autonym="English" data-language-local-name="englanti" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Gertakizun_osagarri" title="Gertakizun osagarri — baski" lang="eu" hreflang="eu" data-title="Gertakizun osagarri" data-language-autonym="Euskara" data-language-local-name="baski" 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_%D9%85%DA%A9%D9%85%D9%84" title="پیشامد مکمل — persia" lang="fa" hreflang="fa" data-title="پیشامد مکمل" data-language-autonym="فارسی" data-language-local-name="persia" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%97%AC%EC%82%AC%EA%B1%B4" title="여사건 — korea" lang="ko" hreflang="ko" data-title="여사건" data-language-autonym="한국어" data-language-local-name="korea" class="interlanguage-link-target"><span>한국어</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_%D7%9E%D7%A9%D7%9C%D7%99%D7%9D" title="מאורע משלים — heprea" lang="he" hreflang="he" data-title="מאורע משלים" data-language-autonym="עברית" data-language-local-name="heprea" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Evento_complementar" title="Evento complementar — portugali" lang="pt" hreflang="pt" data-title="Evento complementar" data-language-autonym="Português" data-language-local-name="portugali" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/T%C3%BCmleyen_olay" title="Tümleyen olay — turkki" lang="tr" hreflang="tr" data-title="Tümleyen olay" data-language-autonym="Türkçe" data-language-local-name="turkki" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%B0%8D%E7%AB%8B%E4%BA%8B%E4%BB%B6" title="對立事件 — kantoninkiina" lang="yue" hreflang="yue" data-title="對立事件" data-language-autonym="粵語" data-language-local-name="kantoninkiina" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q3755832#sitelinks-wikipedia" title="Muokkaa kieltenvälisiä linkkejä" class="wbc-editpage">Muokkaa linkkejä</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Nimiavaruudet"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Vastatapahtuma" title="Näytä sisältösivu [c]" accesskey="c"><span>Artikkeli</span></a></li><li id="ca-talk" class="new vector-tab-noicon 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versiot [h]" accesskey="h"><span>Näytä historia</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Sivutyökalut"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Työkalut" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Työkalut</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Työkalut</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">siirrä sivupalkkiin</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">piilota</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Lisää valintoja" > <div class="vector-menu-heading"> Toiminnot </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Vastatapahtuma"><span>Lue</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Vastatapahtuma&veaction=edit" title="Muokkaa tätä sivua [v]" accesskey="v"><span>Muokkaa</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Vastatapahtuma&action=edit" title="Muokkaa tämän sivun lähdekoodia [e]" accesskey="e"><span>Muokkaa wikitekstiä</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Vastatapahtuma&action=history"><span>Näytä historia</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Yleinen </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Toiminnot:T%C3%A4nne_viittaavat_sivut/Vastatapahtuma" title="Lista sivuista, jotka viittaavat tänne [j]" accesskey="j"><span>Tänne viittaavat sivut</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Toiminnot:Linkitetyt_muutokset/Vastatapahtuma" rel="nofollow" title="Viimeisimmät muokkaukset sivuissa, joille viitataan tältä sivulta [k]" accesskey="k"><span>Linkitettyjen sivujen muutokset</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Vastatapahtuma&oldid=22985189" title="Ikilinkki tämän sivun tähän versioon"><span>Ikilinkki</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Vastatapahtuma&action=info" title="Enemmän tietoa tästä sivusta"><span>Sivun tiedot</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Toiminnot:Viittaus&page=Vastatapahtuma&id=22985189&wpFormIdentifier=titleform" title="Tietoa tämän sivun lainaamisesta"><span>Viitetiedot</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Toiminnot:UrlShortener&url=https%3A%2F%2Ffi.wikipedia.org%2Fwiki%2FVastatapahtuma"><span>Lyhennä URL-osoite</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Toiminnot:QrCode&url=https%3A%2F%2Ffi.wikipedia.org%2Fwiki%2FVastatapahtuma"><span>Lataa QR-koodi</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Tulosta/vie </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Toiminnot:DownloadAsPdf&page=Vastatapahtuma&action=show-download-screen"><span>Lataa PDF-tiedostona</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Vastatapahtuma&printable=yes" title="Tulostettava versio [p]" accesskey="p"><span>Tulostettava versio</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> Muissa hankkeissa </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q3755832" title="Linkki yhdistettyyn keskustietovaraston kohteeseen [g]" accesskey="g"><span>Wikidata-kohde</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Sivutyökalut"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Ulkoasu"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Ulkoasu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">siirrä sivupalkkiin</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">piilota</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Wikipediasta</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="fi" dir="ltr"><p><b>Vastatapahtuma</b> eli <b>komplementtitapahtuma</b> <sup id="cite_ref-ala3_1-0" class="reference"><a href="#cite_note-ala3-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> (<a href="/wiki/Englannin_kieli" title="Englannin kieli">engl.</a> <span lang="en"><i>complementary event</i></span>) tai joskus vain <b>vastatapaus</b> <sup id="cite_ref-hr_2-0" class="reference"><a href="#cite_note-hr-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> on yksi <a href="/wiki/Todenn%C3%A4k%C3%B6isyyslaskenta" class="mw-redirect" title="Todennäköisyyslaskenta">todennäköisyyslaskennassa</a> peruskäsitteistä. Annetun <a href="/wiki/Tapahtuma_(todenn%C3%A4k%C3%B6isyys)" title="Tapahtuma (todennäköisyys)">tapahtuman</a> vastatapahtumalla tarkoitetaan tilannetta, jossa kyseinen tapahtuma ei toteudu. Jos jonkin tapahtuma todennäköisyys on A, sen vastatapahtuman todennäköisyys on 1 – A. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Joukko-opillinen_määritelmä"><span id="Joukko-opillinen_m.C3.A4.C3.A4ritelm.C3.A4"></span>Joukko-opillinen määritelmä</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vastatapahtuma&veaction=edit&section=1" title="Muokkaa osiota Joukko-opillinen määritelmä" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vastatapahtuma&action=edit&section=1" title="Muokkaa osion lähdekoodia: Joukko-opillinen määritelmä"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Jokainen kysymykseen tuleva tapahtuma, jolle todennäköisyys voidaan määrittää, koostuu tietystä joukosta <a href="/wiki/Satunnaisilmi%C3%B6" class="mw-redirect" title="Satunnaisilmiö">satunnaisilmiön</a> <a href="/wiki/Alkeistapaus" title="Alkeistapaus">alkeistapauksia</a>, ja vastatapahtuma muodostuu satunnaisilmiön <a href="/wiki/Perusjoukko_(todenn%C3%A4k%C3%B6isyys)" title="Perusjoukko (todennäköisyys)">perusjoukon</a> kaikista muista alkeistapauksista. Tapahtuman <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> vastatapahtumaa merkitään yleensä <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 {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c003b99422260ba7cc644d36c04448c181a3759" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.676ex;" alt="{\displaystyle {\bar {A}}}" /></span> <sup id="cite_ref-ala3_1-1" class="reference"><a href="#cite_note-ala3-1"><span class="cite-bracket">[</span>1<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"> <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>, <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/7815638370f616885fa8162b06697d078459f5ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.687ex; height:2.343ex;" alt="{\displaystyle A^{c}}" /></span> <sup id="cite_ref-hr_2-1" class="reference"><a href="#cite_note-hr-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-sottinen_3-0" class="reference"><a href="#cite_note-sottinen-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> tai <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^{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82371f50deacf74ee8b65c165272db0f581b3ff9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.263ex; height:2.676ex;" alt="{\displaystyle C^{A}}" /></span> <sup id="cite_ref-kivela1_4-0" class="reference"><a href="#cite_note-kivela1-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>.<sup id="cite_ref-ala3_1-2" class="reference"><a href="#cite_note-ala3-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-jyu33_5-0" class="reference"><a href="#cite_note-jyu33-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-jyu34_6-0" class="reference"><a href="#cite_note-jyu34-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-hr_2-2" class="reference"><a href="#cite_note-hr-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Event_7-0" class="reference"><a href="#cite_note-Event-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-etalukio_8-0" class="reference"><a href="#cite_note-etalukio-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>Vastatapahtuma voidaan merkitä <a href="/wiki/Joukko-oppi" title="Joukko-oppi">joukko-opin</a> käsittein <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 {A}}=\Omega \setminus A.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi mathvariant="normal">Ω</mi> <mo class="MJX-variant">∖</mo> <mi>A</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {A}}=\Omega \setminus A.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bc22bb78827921193db18b51961bf261bcec8cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.137ex; height:3.176ex;" alt="{\displaystyle {\bar {A}}=\Omega \setminus A.}" /></span> Määritelmä tekee tapahtumasta ja vastatapahtumasta <a href="/wiki/Erilliset_tapahtumat" title="Erilliset tapahtumat">erilliset tapahtumat</a>.<sup id="cite_ref-kivela1_4-1" class="reference"><a href="#cite_note-kivela1-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Esimerkkejä"><span id="Esimerkkej.C3.A4"></span>Esimerkkejä</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vastatapahtuma&veaction=edit&section=2" title="Muokkaa osiota Esimerkkejä" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vastatapahtuma&action=edit&section=2" title="Muokkaa osion lähdekoodia: Esimerkkejä"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Tapahtuman alkeistapauksia voidaan kohdella joukon alkioina. Joukot, jotka jakavat perusjoukon kahteen <a href="/wiki/Ositus" title="Ositus">osaan</a>, ovat toisilleen <a href="/wiki/Komplementti_(joukko-oppi)" title="Komplementti (joukko-oppi)">komplementit joukot</a>. Joukon <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>komplementtijoukko <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/7815638370f616885fa8162b06697d078459f5ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.687ex; height:2.343ex;" alt="{\displaystyle A^{c}}" /></span> määritellään <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}:=\{\omega \in \Omega ;\omega \notin A\}.}"> <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> <mo fence="false" stretchy="false">{</mo> <mi>ω</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>;</mo> <mi>ω</mi> <mo>∉</mo> <mi>A</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{c}:=\{\omega \in \Omega ;\omega \notin A\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/401ed862dca9f201e247c1f6b394e13dcd65b7ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.432ex; height:2.843ex;" alt="{\displaystyle A^{c}:=\{\omega \in \Omega ;\omega \notin A\}.}" /></span> Jos tapahtuma on tyhjä joukko, on sen vastatapahtuma perusjoukko, ja päinvastoin: <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 ^{c}=\emptyset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>=</mo> <mi mathvariant="normal">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega ^{c}=\emptyset }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c46b4fcd7a875b004943a2a6c1c950e0f776f78b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.883ex; height:2.509ex;" alt="{\displaystyle \Omega ^{c}=\emptyset }" /></span> ja vastaavasti <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 \emptyset ^{c}=\Omega .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∅</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>=</mo> <mi mathvariant="normal">Ω</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \emptyset ^{c}=\Omega .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f52332720183f2d7fe23fbb09369d344df220e6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.53ex; height:2.509ex;" alt="{\displaystyle \emptyset ^{c}=\Omega .}" /></span> Nopanheitossa tapahtuman "vähintään nelonen" vastatapahtuma olisi "korkeintaan kolmonen" eli <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 \{1,2,3\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2,3\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/800a83c311f50e2f0aa21d0027dd98bd880f72bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.527ex; height:2.843ex;" alt="{\displaystyle \{1,2,3\}.}" /></span> Tapahtuma ja vastatapahtuma muodostavatkin yhdessä perusjoukon: <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 \{1,2,3\}\cup \{4,5,6\}=\{1,2,3,4,5,6\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>6</mn> <mo fence="false" stretchy="false">}</mo> <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 \{1,2,3\}\cup \{4,5,6\}=\{1,2,3,4,5,6\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2f9bda19266c453118fa14ad01a8cce89712566" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.911ex; height:2.843ex;" alt="{\displaystyle \{1,2,3\}\cup \{4,5,6\}=\{1,2,3,4,5,6\}}" /></span> luonnollisella tavalla.<sup id="cite_ref-hr_2-3" class="reference"><a href="#cite_note-hr-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-kivela1_4-2" class="reference"><a href="#cite_note-kivela1-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-sottinen_3-1" class="reference"><a href="#cite_note-sottinen-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Komplementtisääntö"><span id="Komplementtis.C3.A4.C3.A4nt.C3.B6"></span>Komplementtisääntö</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vastatapahtuma&veaction=edit&section=3" title="Muokkaa osiota Komplementtisääntö" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vastatapahtuma&action=edit&section=3" title="Muokkaa osion lähdekoodia: Komplementtisääntö"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Todennäköisyys, että alkeistapaus <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>ω</mi> </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/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" /></span> kuuluu perusjoukkoon <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> </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/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" /></span> on yksi. Samasta syystä voidaan sanoa, että alkeistapaus kuuluu varmuudella aina tapahtumaan tai vastatapahtumaan, sillä perusjoukko muodostuu niistä. Silloin on </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{\text{ tai }}{\bar {A}})=P(A\cup {\bar {A}})=P(A)+P({\bar {A}})=1.}"> <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"> <mtext> tai </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(A{\text{ tai }}{\bar {A}})=P(A\cup {\bar {A}})=P(A)+P({\bar {A}})=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daacfc28c6f89a7a9d7306ff00683db325dea40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.178ex; height:3.176ex;" alt="{\displaystyle P(A{\text{ tai }}{\bar {A}})=P(A\cup {\bar {A}})=P(A)+P({\bar {A}})=1.}" /></span> <sup id="cite_ref-ala3_1-3" class="reference"><a href="#cite_note-ala3-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></dd></dl> <p>Silloin vastatapahtuman <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 {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c003b99422260ba7cc644d36c04448c181a3759" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.676ex;" alt="{\displaystyle {\bar {A}}}" /></span> todennäköisyys on </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({\bar {A}})=1-P(A).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P({\bar {A}})=1-P(A).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08628ed1ec07ac068deb789fc2067412217dcd08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.376ex; height:3.176ex;" alt="{\displaystyle P({\bar {A}})=1-P(A).}" /></span> (<i>komplementtisääntö</i>) <sup id="cite_ref-ala3_1-4" class="reference"><a href="#cite_note-ala3-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-jyu34_6-1" class="reference"><a href="#cite_note-jyu34-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-hr_2-4" class="reference"><a href="#cite_note-hr-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Lähteet"><span id="L.C3.A4hteet"></span>Lähteet</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vastatapahtuma&veaction=edit&section=4" title="Muokkaa osiota Lähteet" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vastatapahtuma&action=edit&section=4" title="Muokkaa osion lähdekoodia: Lähteet"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r23091323">.mw-parser-output div.viitteet-malline ol.references{list-style-type:inherit}.mw-parser-output .viitteet-sarakkeet li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output ol.references>li:target{background-color:var(--background-color-interactive,#DEF)}.mw-parser-output sup.reference:target{background-color:var(--background-color-interactive,#DEF)}.mw-parser-output span[rel="mw:referencedBy"]{counter-reset:mw-ref-linkback 0}.mw-parser-output span[rel="mw:referencedBy"]>a::before{font-style:italic;content:counter(mw-ref-linkback,lower-alpha)}body.action-info .mw-parser-output :target{background-color:var(--background-color-interactive,#DEF)}</style><div id="viitteet-malline" class="viitteet-malline" style="list-style-type:decimal;"><ol class="references"> <li id="cite_note-ala3-1"><span class="mw-cite-backlink">↑ <a href="#cite_ref-ala3_1-0"><sup><i>a</i></sup></a> <a href="#cite_ref-ala3_1-1"><sup><i>b</i></sup></a> <a href="#cite_ref-ala3_1-2"><sup><i>c</i></sup></a> <a href="#cite_ref-ala3_1-3"><sup><i>d</i></sup></a> <a href="#cite_ref-ala3_1-4"><sup><i>e</i></sup></a></span> <span class="reference-text"><span class="kirjaviite" title="Kirjaviite">Alatupa, Sami et al.: <i>Pitkä Sigma 6</i>, s. 110−118.  (lukion pitkän matematiikan oppikirja)  Helsinki:  Tammi, 2010.  <a href="/wiki/Toiminnot:Kirjal%C3%A4hteet/978-951-31-5343-4" title="Toiminnot:Kirjalähteet/978-951-31-5343-4">ISBN 978-951-31-5343-4</a> </span></span> </li> <li id="cite_note-hr-2"><span class="mw-cite-backlink">↑ <a href="#cite_ref-hr_2-0"><sup><i>a</i></sup></a> <a href="#cite_ref-hr_2-1"><sup><i>b</i></sup></a> <a href="#cite_ref-hr_2-2"><sup><i>c</i></sup></a> <a href="#cite_ref-hr_2-3"><sup><i>d</i></sup></a> <a href="#cite_ref-hr_2-4"><sup><i>e</i></sup></a></span> <span class="reference-text">Ruskeapää, Heikki: <a rel="nofollow" class="external text" href="http://users.utu.fi/semet/tod2/TnI.pdf">Todennäköisyyslaskenta I</a> (<a rel="nofollow" class="external text" href="https://web.archive.org/web/20150615011701/http://users.utu.fi/semet/tod2/TnI.pdf">Arkistoitu</a> – Internet Archive)(luentomoniste), Turun Yliopisto, 2012</span> </li> <li id="cite_note-sottinen-3"><span class="mw-cite-backlink">↑ <a href="#cite_ref-sottinen_3-0"><sup><i>a</i></sup></a> <a href="#cite_ref-sottinen_3-1"><sup><i>b</i></sup></a></span> <span class="reference-text">Sottinen, Tommi: <a rel="nofollow" class="external text" href="http://mathstat.helsinki.fi/~tsottine/tnt/">Todennäköisyysteoria, syksy 2006 (10 op, 5 ov)</a> (<a rel="nofollow" class="external text" href="https://web.archive.org/web/20150615093752/http://mathstat.helsinki.fi/~tsottine/tnt/">Arkistoitu</a> – Internet Archive) (luentomoniste), s. 4−13, Helsingin Yliopisto, 2006</span> </li> <li id="cite_note-kivela1-4"><span class="mw-cite-backlink">↑ <a href="#cite_ref-kivela1_4-0"><sup><i>a</i></sup></a> <a href="#cite_ref-kivela1_4-1"><sup><i>b</i></sup></a> <a href="#cite_ref-kivela1_4-2"><sup><i>c</i></sup></a></span> <span class="reference-text">Kivelä, Simo K.: <a rel="nofollow" class="external text" href="https://matta.hut.fi/matta2/isom/html/todnakl1.html">Todennäköisyyslaskennan peruskäsitteet</a>, M niin kuin matematiikka, 10.8.2000</span> </li> <li id="cite_note-jyu33-5"><span class="mw-cite-backlink"><a href="#cite_ref-jyu33_5-0">↑</a></span> <span class="reference-text">Jyväskylän yliopisto: <a rel="nofollow" class="external text" href="http://www.math.jyu.fi/matpo/kirja/dam/index-33.html">VI.2. Äärellinen todennäköisyyskenttä</a>, 2008</span> </li> <li id="cite_note-jyu34-6"><span class="mw-cite-backlink">↑ <a href="#cite_ref-jyu34_6-0"><sup><i>a</i></sup></a> <a href="#cite_ref-jyu34_6-1"><sup><i>b</i></sup></a></span> <span class="reference-text">Jyväskylän yliopisto: <a rel="nofollow" class="external text" href="http://www.math.jyu.fi/matpo/kirja/dam/index-34.html">VI.2. Aksiomatisointi</a>, 2008</span> </li> <li id="cite_note-Event-7"><span class="mw-cite-backlink"><a href="#cite_ref-Event_7-0">↑</a></span> <span class="reference-text"><span class="verkkoviite" title="Verkkoviite">Weisstein, Eric W.: <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Event.html">Event</a> (Math World – A Wolfram Web Resource) Wolfram Research. <span style="font-size: 0.95em; position: relative;">(englanniksi)</span></span></span> </li> <li id="cite_note-etalukio-8"><span class="mw-cite-backlink"><a href="#cite_ref-etalukio_8-0">↑</a></span> <span class="reference-text">Etälukio: <a rel="nofollow" class="external text" href="http://www02.oph.fi/etalukio/pitka_matematiikka/kurssi6/maa6_teoria6.html">Komplementtitapaus</a> (<a rel="nofollow" class="external text" href="https://web.archive.org/web/20151128052836/http://www02.oph.fi/etalukio/pitka_matematiikka/kurssi6/maa6_teoria6.html">Arkistoitu</a> – Internet Archive)</span> </li> </ol> </div></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Noudettu kohteesta ”<a dir="ltr" href="https://fi.wikipedia.org/w/index.php?title=Vastatapahtuma&oldid=22985189">https://fi.wikipedia.org/w/index.php?title=Vastatapahtuma&oldid=22985189</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Toiminnot:Luokat" title="Toiminnot:Luokat">Luokka</a>: <ul><li><a href="/wiki/Luokka:Todenn%C3%A4k%C3%B6isyyslaskenta" title="Luokka:Todennäköisyyslaskenta">Todennäköisyyslaskenta</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Sivua on viimeksi muutettu 27. tammikuuta 2025 kello 20.44.</li> <li id="footer-info-copyright">Teksti on saatavilla <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.fi">Creative Commons Attribution/Share-Alike</a> -lisenssillä; lisäehtoja voi sisältyä. 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