Hypergeometric distribution

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<button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Properties subsection</span> </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Working_example" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Working_example"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Working example</span> </div> </a> <ul id="toc-Working_example-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Symmetries" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Symmetries"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Symmetries</span> </div> </a> <ul id="toc-Symmetries-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Order_of_draws" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Order_of_draws"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Order of draws</span> </div> </a> <ul id="toc-Order_of_draws-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tail_bounds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tail_bounds"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Tail bounds</span> </div> </a> <ul id="toc-Tail_bounds-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Statistical_Inference" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Statistical_Inference"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Statistical Inference</span> </div> </a> <button aria-controls="toc-Statistical_Inference-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Statistical Inference subsection</span> </button> <ul id="toc-Statistical_Inference-sublist" class="vector-toc-list"> <li id="toc-Hypergeometric_test" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hypergeometric_test"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Hypergeometric test</span> </div> </a> <ul id="toc-Hypergeometric_test-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Related_distributions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Related_distributions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Related distributions</span> </div> </a> <button aria-controls="toc-Related_distributions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Related distributions subsection</span> </button> <ul id="toc-Related_distributions-sublist" class="vector-toc-list"> <li id="toc-Multivariate_hypergeometric_distribution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multivariate_hypergeometric_distribution"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Multivariate hypergeometric distribution</span> </div> </a> <ul id="toc-Multivariate_hypergeometric_distribution-sublist" class="vector-toc-list"> <li id="toc-Example" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.1</span> <span>Example</span> </div> </a> <ul id="toc-Example-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Occurrence_and_applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Occurrence_and_applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Occurrence and applications</span> </div> </a> <button aria-controls="toc-Occurrence_and_applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Occurrence and applications subsection</span> </button> <ul id="toc-Occurrence_and_applications-sublist" class="vector-toc-list"> <li id="toc-Application_to_auditing_elections" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Application_to_auditing_elections"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Application to auditing elections</span> </div> </a> <ul id="toc-Application_to_auditing_elections-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Application_to_Texas_hold'em_poker" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Application_to_Texas_hold'em_poker"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Application to Texas hold'em poker</span> </div> </a> <ul id="toc-Application_to_Texas_hold'em_poker-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Application_to_Keno" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Application_to_Keno"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Application to Keno</span> </div> </a> <ul id="toc-Application_to_Keno-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Citations</span> </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>External links</span> </div> </a> <ul id="toc-External_links-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="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <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">Toggle the table of contents</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">Hypergeometric distribution</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" 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Available in 30 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-30" 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">30 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%93%D1%96%D0%BF%D0%B5%D1%80%D0%B3%D0%B5%D0%B0%D0%BC%D0%B5%D1%82%D1%80%D1%8B%D1%87%D0%BD%D0%B0%D0%B5_%D1%80%D0%B0%D0%B7%D0%BC%D0%B5%D1%80%D0%BA%D0%B0%D0%B2%D0%B0%D0%BD%D0%BD%D0%B5" title="Гіпергеаметрычнае размеркаванне – Belarusian" lang="be" hreflang="be" data-title="Гіпергеаметрычнае размеркаванне" 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/Distribuci%C3%B3_hipergeom%C3%A8trica" title="Distribució hipergeomètrica – Catalan" lang="ca" hreflang="ca" data-title="Distribució hipergeomètrica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Hypergeometrick%C3%A9_rozd%C4%9Blen%C3%AD" title="Hypergeometrické rozdělení – Czech" lang="cs" hreflang="cs" data-title="Hypergeometrické rozdělení" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Hypergeometrische_Verteilung" title="Hypergeometrische Verteilung – German" lang="de" hreflang="de" data-title="Hypergeometrische Verteilung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A5%CF%80%CE%B5%CF%81%CE%B3%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%B9%CE%BA%CE%AE_%CE%BA%CE%B1%CF%84%CE%B1%CE%BD%CE%BF%CE%BC%CE%AE" title="Υπεργεωμετρική κατανομή – Greek" lang="el" hreflang="el" data-title="Υπεργεωμετρική κατανομή" 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/Distribuci%C3%B3n_hipergeom%C3%A9trica" title="Distribución hipergeométrica – Spanish" lang="es" hreflang="es" data-title="Distribución hipergeométrica" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Banaketa_hipergeometriko" title="Banaketa hipergeometriko – Basque" lang="eu" hreflang="eu" data-title="Banaketa hipergeometriko" 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/%D8%AA%D9%88%D8%B2%DB%8C%D8%B9_%D9%81%D9%88%D9%82%E2%80%8C%D9%87%D9%86%D8%AF%D8%B3%DB%8C" title="توزیع فوق‌هندسی – Persian" lang="fa" hreflang="fa" data-title="توزیع فوق‌هندسی" 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/Loi_hyperg%C3%A9om%C3%A9trique" title="Loi hypergéométrique – French" lang="fr" hreflang="fr" data-title="Loi hypergéométrique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Distribuci%C3%B3n_hiperxeom%C3%A9trica" title="Distribución hiperxeométrica – Galician" lang="gl" hreflang="gl" data-title="Distribución hiperxeométrica" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%B4%88%EA%B8%B0%ED%95%98_%EB%B6%84%ED%8F%AC" title="초기하 분포 – Korean" lang="ko" hreflang="ko" data-title="초기하 분포" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Distribuzione_ipergeometrica" title="Distribuzione ipergeometrica – Italian" lang="it" hreflang="it" data-title="Distribuzione ipergeometrica" 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%94%D7%AA%D7%A4%D7%9C%D7%92%D7%95%D7%AA_%D7%94%D7%99%D7%A4%D7%A8%D7%92%D7%90%D7%95%D7%9E%D7%98%D7%A8%D7%99%D7%AA" title="התפלגות היפרגאומטרית – Hebrew" lang="he" hreflang="he" data-title="התפלגות היפרגאומטרית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Hipergeometrikus_eloszl%C3%A1s" title="Hipergeometrikus eloszlás – Hungarian" lang="hu" hreflang="hu" data-title="Hipergeometrikus eloszlás" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A5%D0%B8%D0%BF%D0%B5%D1%80%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%81%D0%BA%D0%B0_%D1%80%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B1%D0%B0" title="Хипергеометриска распределба – Macedonian" lang="mk" hreflang="mk" data-title="Хипергеометриска распределба" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Hypergeometrische_verdeling" title="Hypergeometrische verdeling – Dutch" lang="nl" hreflang="nl" data-title="Hypergeometrische verdeling" 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/%E8%B6%85%E5%B9%BE%E4%BD%95%E5%88%86%E5%B8%83" title="超幾何分布 – Japanese" lang="ja" hreflang="ja" data-title="超幾何分布" 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/Hypergeometrisk_fordeling" title="Hypergeometrisk fordeling – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Hypergeometrisk fordeling" 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-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Hypergeometrisk_fordeling" title="Hypergeometrisk fordeling – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Hypergeometrisk fordeling" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Rozk%C5%82ad_hipergeometryczny" title="Rozkład hipergeometryczny – Polish" lang="pl" hreflang="pl" data-title="Rozkład hipergeometryczny" 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/Distribui%C3%A7%C3%A3o_hipergeom%C3%A9trica" title="Distribuição hipergeométrica – Portuguese" lang="pt" hreflang="pt" data-title="Distribuição hipergeométrica" 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%93%D0%B8%D0%BF%D0%B5%D1%80%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D1%80%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5" title="Гипергеометрическое распределение – Russian" lang="ru" hreflang="ru" data-title="Гипергеометрическое распределение" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Shp%C3%ABrndarja_hipergjeometrike" title="Shpërndarja hipergjeometrike – Albanian" lang="sq" hreflang="sq" data-title="Shpërndarja hipergjeometrike" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Hipergeometri%C4%8Dna_porazdelitev" title="Hipergeometrična porazdelitev – Slovenian" lang="sl" hreflang="sl" data-title="Hipergeometrična porazdelitev" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Hypergeometrinen_jakauma" title="Hypergeometrinen jakauma – Finnish" lang="fi" hreflang="fi" data-title="Hypergeometrinen jakauma" 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/Hypergeometrisk_f%C3%B6rdelning" title="Hypergeometrisk fördelning – Swedish" lang="sv" hreflang="sv" data-title="Hypergeometrisk fördelning" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Distribusyong_hiperheometriko" title="Distribusyong hiperheometriko – Tagalog" lang="tl" hreflang="tl" data-title="Distribusyong hiperheometriko" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Hipergeometrik_da%C4%9F%C4%B1l%C4%B1m" title="Hipergeometrik dağılım – Turkish" lang="tr" hreflang="tr" data-title="Hipergeometrik dağılım" 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 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id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Discrete probability distribution</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Geometric_distribution" title="Geometric distribution">Geometric distribution</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><style data-mw-deduplicate="TemplateStyles:r1247679731">.mw-parser-output .ib-prob-dist{border-collapse:collapse;width:20em}.mw-parser-output .ib-prob-dist td,.mw-parser-output .ib-prob-dist th{border:1px solid var(--border-color-base,#a2a9b1)}.mw-parser-output .ib-prob-dist .infobox-subheader{text-align:left}.mw-parser-output .ib-prob-dist-image{background:var(--background-color-neutral,#eaecf0);font-weight:bold;text-align:center}</style><table class="infobox infobox-table ib-prob-dist"><caption class="infobox-title">Hypergeometric</caption><tbody><tr><td colspan="4" class="infobox-image"> <div class="ib-prob-dist-image">Probability mass function</div><span typeof="mw:File"><a href="/wiki/File:HypergeometricPDF.png" class="mw-file-description" title="Hypergeometric PDF plot"><img alt="Hypergeometric PDF plot" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/HypergeometricPDF.png/300px-HypergeometricPDF.png" decoding="async" width="300" height="257" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/HypergeometricPDF.png/450px-HypergeometricPDF.png 1.5x, //upload.wikimedia.org/wikipedia/commons/c/c1/HypergeometricPDF.png 2x" data-file-width="515" data-file-height="442" /></a></span></td></tr><tr><td colspan="4" class="infobox-image"> <div class="ib-prob-dist-image">Cumulative distribution function</div><span typeof="mw:File"><a href="/wiki/File:HypergeometricCDF.png" class="mw-file-description" title="Hypergeometric CDF plot"><img alt="Hypergeometric CDF plot" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/HypergeometricCDF.png/300px-HypergeometricCDF.png" decoding="async" width="300" height="257" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/HypergeometricCDF.png/450px-HypergeometricCDF.png 1.5x, //upload.wikimedia.org/wikipedia/commons/b/b4/HypergeometricCDF.png 2x" data-file-width="515" data-file-height="442" /></a></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Statistical_parameter" title="Statistical parameter">Parameters</a></th><td colspan="3" class="infobox-data"> <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 {\begin{aligned}N&\in \left\{0,1,2,\dots \right\}\\K&\in \left\{0,1,2,\dots ,N\right\}\\n&\in \left\{0,1,2,\dots ,N\right\}\end{aligned}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>N</mi> </mtd> <mtd> <mi></mi> <mo>∈</mo> <mrow> <mo>{</mo> <mrow> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>K</mi> </mtd> <mtd> <mi></mi> <mo>∈</mo> <mrow> <mo>{</mo> <mrow> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>N</mi> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>n</mi> </mtd> <mtd> <mi></mi> <mo>∈</mo> <mrow> <mo>{</mo> <mrow> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>N</mi> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> </mtable> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}N&\in \left\{0,1,2,\dots \right\}\\K&\in \left\{0,1,2,\dots ,N\right\}\\n&\in \left\{0,1,2,\dots ,N\right\}\end{aligned}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7111c6500f27abdf871de43655e68f6cb8cccfe5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:21.167ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}N&\in \left\{0,1,2,\dots \right\}\\K&\in \left\{0,1,2,\dots ,N\right\}\\n&\in \left\{0,1,2,\dots ,N\right\}\end{aligned}}\,}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Support_(mathematics)" title="Support (mathematics)">Support</a></th><td colspan="3" class="infobox-data"> <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 \scriptstyle {k\,\in \,\left\{\max {(0,\,n+K-N)},\,\dots ,\,\min {(n,\,K)}\right\}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mspace width="thinmathspace" /> <mo>∈</mo> <mspace width="thinmathspace" /> <mrow> <mo>{</mo> <mrow> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mi>n</mi> <mo>+</mo> <mi>K</mi> <mo>−</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mo>…</mo> <mo>,</mo> <mspace width="thinmathspace" /> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mo>}</mo> </mrow> </mrow> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {k\,\in \,\left\{\max {(0,\,n+K-N)},\,\dots ,\,\min {(n,\,K)}\right\}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30944d7303c29e3aef3ddf88b62e5d2d18f3cb25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.926ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {k\,\in \,\left\{\max {(0,\,n+K-N)},\,\dots ,\,\min {(n,\,K)}\right\}}\,}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Probability_mass_function" title="Probability mass function">PMF</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {{{K \choose k}{{N-K} \choose {n-k}}} \over {N \choose n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>K</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mi>K</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>N</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {{{K \choose k}{{N-K} \choose {n-k}}} \over {N \choose n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e9b0e44abfe907585b625d3b15466067e1a4917" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:10.755ex; height:7.843ex;" alt="{\displaystyle {{{K \choose k}{{N-K} \choose {n-k}}} \over {N \choose n}}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">CDF</a></th><td colspan="3" class="infobox-data"> <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-{{{n \choose {k+1}}{{N-n} \choose {K-k-1}}} \over {N \choose K}}\,_{3}F_{2}\!\!\left[{\begin{array}{c}1,\ k+1-K,\ k+1-n\\k+2,\ N+k+2-K-n\end{array}};1\right],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mo>−</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>N</mi> <mi>K</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mfrac> </mrow> <msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>K</mi> <mo>,</mo> <mtext> </mtext> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo>,</mo> <mtext> </mtext> <mi>N</mi> <mo>+</mo> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo>−</mo> <mi>K</mi> <mo>−</mo> <mi>n</mi> </mtd> </mtr> </mtable> </mrow> <mo>;</mo> <mn>1</mn> </mrow> <mo>]</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-{{{n \choose {k+1}}{{N-n} \choose {K-k-1}}} \over {N \choose K}}\,_{3}F_{2}\!\!\left[{\begin{array}{c}1,\ k+1-K,\ k+1-n\\k+2,\ N+k+2-K-n\end{array}};1\right],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/556153b7446bc4321c995b2d8f9cc2957a0df452" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:53.88ex; height:7.843ex;" alt="{\displaystyle 1-{{{n \choose {k+1}}{{N-n} \choose {K-k-1}}} \over {N \choose K}}\,_{3}F_{2}\!\!\left[{\begin{array}{c}1,\ k+1-K,\ k+1-n\\k+2,\ N+k+2-K-n\end{array}};1\right],}"></span> where <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}F_{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,_{p}F_{q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c5609cf9098e640e342465c1d255923c0f8b6c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.929ex; height:2.843ex;" alt="{\displaystyle \,_{p}F_{q}}"></span> is the <a href="/wiki/Generalized_hypergeometric_function" title="Generalized hypergeometric function">generalized hypergeometric function</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Expected_value" title="Expected value">Mean</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n{K \over N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>K</mi> <mi>N</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n{K \over N}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f058199d005892337737985dd87d62ada13fc93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.297ex; height:5.176ex;" alt="{\displaystyle n{K \over N}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Mode_(statistics)" title="Mode (statistics)">Mode</a></th><td colspan="3" class="infobox-data"> <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 \left\lceil {\frac {(n+1)(K+1)}{N+2}}\right\rceil -1,\left\lfloor {\frac {(n+1)(K+1)}{N+2}}\right\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⌈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>K</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>N</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>⌉</mo> </mrow> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>K</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>N</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>⌋</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\lceil {\frac {(n+1)(K+1)}{N+2}}\right\rceil -1,\left\lfloor {\frac {(n+1)(K+1)}{N+2}}\right\rfloor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/242a00e7177e246df70098dcb184ec9fd41a5b71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.3ex; height:6.343ex;" alt="{\displaystyle \left\lceil {\frac {(n+1)(K+1)}{N+2}}\right\rceil -1,\left\lfloor {\frac {(n+1)(K+1)}{N+2}}\right\rfloor }"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Variance" title="Variance">Variance</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n{K \over N}{N-K \over N}{N-n \over N-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>K</mi> <mi>N</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>N</mi> <mo>−</mo> <mi>K</mi> </mrow> <mi>N</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>N</mi> <mo>−</mo> <mi>n</mi> </mrow> <mrow> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n{K \over N}{N-K \over N}{N-n \over N-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dd69c6e53c01a36ab1ab7480b5cdad03f5b3e5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.238ex; height:5.343ex;" alt="{\displaystyle n{K \over N}{N-K \over N}{N-n \over N-1}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Skewness" title="Skewness">Skewness</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {(N-2K)(N-1)^{\frac {1}{2}}(N-2n)}{[nK(N-K)(N-n)]^{\frac {1}{2}}(N-2)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mn>2</mn> <mi>K</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">[</mo> <mi>n</mi> <mi>K</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {(N-2K)(N-1)^{\frac {1}{2}}(N-2n)}{[nK(N-K)(N-n)]^{\frac {1}{2}}(N-2)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9ec1b0c28225251fa3fd794e30bffc3eb34315e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:32.089ex; height:8.843ex;" alt="{\displaystyle {\frac {(N-2K)(N-1)^{\frac {1}{2}}(N-2n)}{[nK(N-K)(N-n)]^{\frac {1}{2}}(N-2)}}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Excess_kurtosis" class="mw-redirect" title="Excess kurtosis">Excess kurtosis</a></th><td colspan="3" class="infobox-data"> <p><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 \left.{\frac {1}{nK(N-K)(N-n)(N-2)(N-3)}}\cdot \right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mi>K</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>⋅</mo> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.{\frac {1}{nK(N-K)(N-n)(N-2)(N-3)}}\cdot \right.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c6e2651be60e16aa2e5a7a14c4e102850ebb81c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:37.582ex; height:6.009ex;" alt="{\displaystyle \left.{\frac {1}{nK(N-K)(N-n)(N-2)(N-3)}}\cdot \right.}"></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 {\Big [}(N-1)N^{2}{\Big (}N(N+1)-6K(N-K)-6n(N-n){\Big )}+{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mi>N</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>6</mn> <mi>K</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>6</mn> <mi>n</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Big [}(N-1)N^{2}{\Big (}N(N+1)-6K(N-K)-6n(N-n){\Big )}+{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d94b35ebcaa930329e2980f25c90cebc31e72e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:56.059ex; height:4.843ex;" alt="{\displaystyle {\Big [}(N-1)N^{2}{\Big (}N(N+1)-6K(N-K)-6n(N-n){\Big )}+{}}"></span> </p> <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 {}+6nK(N-K)(N-n)(5N-6){\Big ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>+</mo> <mn>6</mn> <mi>n</mi> <mi>K</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>5</mn> <mi>N</mi> <mo>−</mo> <mn>6</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}+6nK(N-K)(N-n)(5N-6){\Big ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046f280d3e6c514b12a25ec51c9159874d811f39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.486ex; height:4.843ex;" alt="{\displaystyle {}+6nK(N-K)(N-n)(5N-6){\Big ]}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Moment-generating_function" title="Moment-generating function">MGF</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{N-K \choose n}\scriptstyle {\,_{2}F_{1}(-n,-K;N-K-n+1;e^{t})}}{N \choose n}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>N</mi> <mo>−</mo> <mi>K</mi> </mrow> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mo>−</mo> <mi>n</mi> <mo>,</mo> <mo>−</mo> <mi>K</mi> <mo>;</mo> <mi>N</mi> <mo>−</mo> <mi>K</mi> <mo>−</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>;</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>N</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{N-K \choose n}\scriptstyle {\,_{2}F_{1}(-n,-K;N-K-n+1;e^{t})}}{N \choose n}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d7f06ec0cd1c3ffc74d5eba315d7a4586e0b66c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; margin-right: -0.387ex; width:28.291ex; height:7.843ex;" alt="{\displaystyle {\frac {{N-K \choose n}\scriptstyle {\,_{2}F_{1}(-n,-K;N-K-n+1;e^{t})}}{N \choose n}}\,\!}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">CF</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{N-K \choose n}\scriptstyle {\,_{2}F_{1}(-n,-K;N-K-n+1;e^{it})}}{N \choose n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>N</mi> <mo>−</mo> <mi>K</mi> </mrow> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mo>−</mo> <mi>n</mi> <mo>,</mo> <mo>−</mo> <mi>K</mi> <mo>;</mo> <mi>N</mi> <mo>−</mo> <mi>K</mi> <mo>−</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>;</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>N</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{N-K \choose n}\scriptstyle {\,_{2}F_{1}(-n,-K;N-K-n+1;e^{it})}}{N \choose n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6e3058b8b61e6d968f720c465ac2707cb27c593" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:28.364ex; height:7.843ex;" alt="{\displaystyle {\frac {{N-K \choose n}\scriptstyle {\,_{2}F_{1}(-n,-K;N-K-n+1;e^{it})}}{N \choose n}}}"></span></td></tr></tbody></table> <p>In <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a> and <a href="/wiki/Statistics" title="Statistics">statistics</a>, the <b>hypergeometric distribution</b> is a <a href="/wiki/Probability_distribution#Discrete_probability_distribution" title="Probability distribution">discrete probability distribution</a> that describes 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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> successes (random draws for which the object drawn has a specified feature) in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> draws, <i>without</i> replacement, from a finite <a href="/wiki/Statistical_population" title="Statistical population">population</a> of size <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> that contains exactly <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> objects with that feature, wherein each draw is either a success or a failure. In contrast, the <a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial distribution</a> describes 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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> successes in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> draws <i>with</i> replacement. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hypergeometric_distribution&action=edit&section=1" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Probability_mass_function">Probability mass function</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hypergeometric_distribution&action=edit&section=2" title="Edit section: Probability mass function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following conditions characterize the hypergeometric distribution: </p> <ul><li>The result of each draw (the elements of the population being sampled) can be classified into one of <a href="/wiki/Binary_variable" class="mw-redirect" title="Binary variable">two mutually exclusive categories</a> (e.g. Pass/Fail or Employed/Unemployed).</li> <li>The probability of a success changes on each draw, as each draw decreases the population (<i><a href="/wiki/Sampling_without_replacement" class="mw-redirect" title="Sampling without replacement">sampling without replacement</a></i> from a finite population).</li></ul> <p>A <a href="/wiki/Random_variable" title="Random variable">random variable</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> follows the hypergeometric distribution if its <a href="/wiki/Probability_mass_function" title="Probability mass function">probability mass function</a> (pmf) is given by<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> </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_{X}(k)=\Pr(X=k)={\frac {{\binom {K}{k}}{\binom {N-K}{n-k}}}{\binom {N}{n}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>=</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>K</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>N</mi> <mo>−</mo> <mi>K</mi> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>N</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{X}(k)=\Pr(X=k)={\frac {{\binom {K}{k}}{\binom {N-K}{n-k}}}{\binom {N}{n}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/928a49d5765a69e6abd87020769049318264c06a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; margin-left: -0.089ex; width:34.104ex; height:7.843ex;" alt="{\displaystyle p_{X}(k)=\Pr(X=k)={\frac {{\binom {K}{k}}{\binom {N-K}{n-k}}}{\binom {N}{n}}},}"></span></dd></dl> <p>where </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> is the population size,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> is the number of success states in the population,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> is the number of draws (i.e. quantity drawn in each trial),</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> is the number of observed successes,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \textstyle {a \choose b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>a</mi> <mi>b</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \textstyle {a \choose b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6622c3cc26bfc1b8d759187ce99942a58a7a623" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.999ex; height:3.176ex;" alt="{\textstyle \textstyle {a \choose b}}"></span> is a <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficient</a>.</li></ul> <p>The <abbr title="probability mass function">pmf</abbr> is positive when <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 \max(0,n+K-N)\leq k\leq \min(K,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>n</mi> <mo>+</mo> <mi>K</mi> <mo>−</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>k</mi> <mo>≤</mo> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max(0,n+K-N)\leq k\leq \min(K,n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/306fe973a01775991c458f55b5ac0c283d37ae95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.124ex; height:2.843ex;" alt="{\displaystyle \max(0,n+K-N)\leq k\leq \min(K,n)}"></span>. </p><p>A random variable distributed hypergeometrically with parameters <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> 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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> is written <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="{\textstyle X\sim \operatorname {Hypergeometric} (N,K,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mi>Hypergeometric</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo>,</mo> <mi>K</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle X\sim \operatorname {Hypergeometric} (N,K,n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92b01d6c5b4a75467d2c7945e2ee94db19f40548" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.508ex; height:2.843ex;" alt="{\textstyle X\sim \operatorname {Hypergeometric} (N,K,n)}"></span> and has <a href="/wiki/Probability_mass_function" title="Probability mass function">probability mass function</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="{\textstyle p_{X}(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle p_{X}(k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bd337147473352969b3c4fe170478c25ac01237" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.912ex; height:2.843ex;" alt="{\textstyle p_{X}(k)}"></span> above. </p> <div class="mw-heading mw-heading3"><h3 id="Combinatorial_identities">Combinatorial identities</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hypergeometric_distribution&action=edit&section=3" title="Edit section: Combinatorial identities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As required, we have </p> <dl><dd><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 \sum _{0\leq k\leq {\textrm {min}}(n,K)}{{K \choose k}{N-K \choose n-k} \over {N \choose n}}=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>≤</mo> <mi>k</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>min</mtext> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>K</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>N</mi> <mo>−</mo> <mi>K</mi> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>N</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{0\leq k\leq {\textrm {min}}(n,K)}{{K \choose k}{N-K \choose n-k} \over {N \choose n}}=1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8a5b7729f3d882689fc5e987a5fd23192f954d0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:27.21ex; height:8.009ex;" alt="{\displaystyle \sum _{0\leq k\leq {\textrm {min}}(n,K)}{{K \choose k}{N-K \choose n-k} \over {N \choose n}}=1,}"></span></dd></dl> <p>which essentially follows from <a href="/wiki/Vandermonde%27s_identity" title="Vandermonde's identity">Vandermonde's identity</a> from <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a>. </p><p>Also note that </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 {{K \choose k}{N-K \choose n-k} \over {N \choose n}}={{{n \choose k}{{N-n} \choose {K-k}}} \over {N \choose K}};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>K</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>N</mi> <mo>−</mo> <mi>K</mi> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>N</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mo>−</mo> <mi>k</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>N</mi> <mi>K</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {{K \choose k}{N-K \choose n-k} \over {N \choose n}}={{{n \choose k}{{N-n} \choose {K-k}}} \over {N \choose K}};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2110c7e038c685914e241cdc7ac0136a1aa3b5b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:24.306ex; height:7.843ex;" alt="{\displaystyle {{K \choose k}{N-K \choose n-k} \over {N \choose n}}={{{n \choose k}{{N-n} \choose {K-k}}} \over {N \choose K}};}"></span></dd></dl> <p>This identity can be shown by expressing the binomial coefficients in terms of factorials and rearranging the latter. Additionally, it follows from the symmetry of the problem, described in two different but interchangeable ways. </p><p>For example, consider two rounds of drawing without replacement. In the first round, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> out 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 N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> neutral marbles are drawn from an urn without replacement and coloured green. Then the colored marbles are put back. In the second round, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> marbles are drawn without replacement and colored red. Then, the number of marbles with both colors on them (that is, the number of marbles that have been drawn twice) has the hypergeometric distribution. The symmetry in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> 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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> stems from the fact that the two rounds are independent, and one could have started by drawing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> balls and colouring them red first. </p><p><i>Note that we are interested in 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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> successes in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> draws <b>without replacement</b>, since the probability of success on each trial is not the same, as the size of the remaining population changes as we remove each marble. Keep in mind not to confuse with the <a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial distribution</a>, which describes 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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> successes in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> draws <b>with replacement.</b></i> </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hypergeometric_distribution&action=edit&section=4" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Working_example">Working example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hypergeometric_distribution&action=edit&section=5" title="Edit section: Working example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The classical application of the hypergeometric distribution is <b>sampling without replacement</b>. Think of an <a href="/wiki/Urn_problem" title="Urn problem">urn</a> with two colors of <a href="/wiki/Marbles" class="mw-redirect" title="Marbles">marbles</a>, red and green. Define drawing a green marble as a success and drawing a red marble as a failure. Let <i>N</i> describe the number of <b>all marbles in the urn</b> (see contingency table below) and <i>K</i> describe the number of <b>green marbles</b>, then <i>N</i> − <i>K</i> corresponds to the number of <b>red marbles</b>. Now, standing next to the urn, you close your eyes and draw n marbles without replacement. Define <i>X</i> as a <a href="/wiki/Random_variable" title="Random variable">random variable</a> whose outcome is <i>k</i>, the number of green marbles drawn in the experiment. This situation is illustrated by the following <a href="/wiki/Contingency_table" title="Contingency table">contingency table</a>: </p> <table class="wikitable" style="text-align:center"> <tbody><tr> <th></th> <th>drawn</th> <th>not drawn</th> <th>total </th></tr> <tr> <td align="right"><b>green marbles</b></td> <td><i>k</i></td> <td><i>K</i> − <i>k</i></td> <td><i>K</i> </td></tr> <tr> <td align="right"><b>red marbles</b></td> <td><i>n</i> − <i>k</i></td> <td><i>N + k − n − K</i></td> <td><i>N − K</i> </td></tr> <tr> <td align="right"><b>total</b></td> <td><i>n</i></td> <td><i>N − n</i></td> <td><i>N</i> </td></tr> </tbody></table> <p>Indeed, we are interested in calculating the probability of drawing k green marbles in n draws, given that there are K green marbles out of a total of N marbles. For this example, assume that there are <b>5</b> green and <b>45</b> red marbles in the urn. Standing next to the urn, you close your eyes and draw <b>10</b> marbles without replacement. What is the probability that exactly <b>4</b> of the <b>10</b> are green? </p><p>This problem is summarized by the following contingency table: </p> <table class="wikitable" style="text-align:center"> <tbody><tr> <th></th> <th>drawn</th> <th>not drawn</th> <th>total </th></tr> <tr> <td align="right"><b>green marbles</b> </td> <td><i>k</i> = <b>4</b> </td> <td><i>K</i> − <i>k</i> = <b>1</b> </td> <td><i>K</i> = <b>5</b> </td></tr> <tr> <td align="right"><b>red marbles</b> </td> <td><i>n</i> − <i>k</i> = <b>6</b> </td> <td><i>N + k − n − K</i> = <b>39</b> </td> <td><i>N − K</i> = <b>45</b> </td></tr> <tr> <td align="right"><b>total</b> </td> <td><i>n</i> = <b>10</b> </td> <td><i>N − n</i> = <b>40</b> </td> <td> </td></tr></tbody></table> <p>To find the probability of <b>drawing k green marbles in exactly n draws out of N total draws</b>, we identify X as a hyper-geometric random variable to use the formula </p><p><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(X=k)=f(k;N,K,n)={{{K \choose k}{{N-K} \choose {n-k}}} \over {N \choose n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>=</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>N</mi> <mo>,</mo> <mi>K</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>K</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mi>K</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>N</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(X=k)=f(k;N,K,n)={{{K \choose k}{{N-K} \choose {n-k}}} \over {N \choose n}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb1f04d4adc49785408d221abb404bfe5751b2c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:40.369ex; height:7.843ex;" alt="{\displaystyle P(X=k)=f(k;N,K,n)={{{K \choose k}{{N-K} \choose {n-k}}} \over {N \choose n}}.}"></span> </p><p>To intuitively explain the given formula, consider the two symmetric problems represented by the identity </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {{K \choose k}{N-K \choose n-k} \over {N \choose n}}={{{n \choose k}{{N-n} \choose {K-k}}} \over {N \choose K}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>K</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>N</mi> <mo>−</mo> <mi>K</mi> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>N</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mo>−</mo> <mi>k</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>N</mi> <mi>K</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {{K \choose k}{N-K \choose n-k} \over {N \choose n}}={{{n \choose k}{{N-n} \choose {K-k}}} \over {N \choose K}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/923d136afd3bf0432a13106154513762b5b90cf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:23.66ex; height:7.843ex;" alt="{\displaystyle {{K \choose k}{N-K \choose n-k} \over {N \choose n}}={{{n \choose k}{{N-n} \choose {K-k}}} \over {N \choose K}}}"></span> </p> <ol><li>left-hand side - drawing a total of only n marbles out of the urn. We want to find the probability of the outcome of <b>drawing k green marbles out of K total green marbles, and drawing n-k red marbles out of N-K red marbles, in these n rounds.</b></li> <li>right hand side - alternatively, drawing all N marbles out of the urn. We want to find the probability of the outcome of drawing <b>k green marbles in n draws out of the total N draws, and K-k green marbles in the rest N-n draws.</b></li></ol> <p>Back to the calculations, we use the formula above to calculate the probability of drawing exactly <i>k</i> green marbles </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(X=4)=f(4;50,5,10)={{{5 \choose 4}{{45} \choose {6}}} \over {50 \choose 10}}={5\cdot 8145060 \over 10272278170}=0.003964583\dots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>=</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo>;</mo> <mn>50</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>10</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>5</mn> <mn>4</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>45</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>50</mn> <mn>10</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>5</mn> <mo>⋅</mo> <mn>8145060</mn> </mrow> <mn>10272278170</mn> </mfrac> </mrow> <mo>=</mo> <mn>0.003964583</mn> <mo>…</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(X=4)=f(4;50,5,10)={{{5 \choose 4}{{45} \choose {6}}} \over {50 \choose 10}}={5\cdot 8145060 \over 10272278170}=0.003964583\dots .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e21ed09f25987e090792ed02ed6be94372ef8de4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:72.954ex; height:7.843ex;" alt="{\displaystyle P(X=4)=f(4;50,5,10)={{{5 \choose 4}{{45} \choose {6}}} \over {50 \choose 10}}={5\cdot 8145060 \over 10272278170}=0.003964583\dots .}"></span></dd></dl> <p>Intuitively we would expect it to be even more unlikely that all 5 green marbles will be among the 10 drawn. </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(X=5)=f(5;50,5,10)={{{5 \choose 5}{{45} \choose {5}}} \over {50 \choose 10}}={1\cdot 1221759 \over 10272278170}=0.0001189375\dots ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>=</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>5</mn> <mo>;</mo> <mn>50</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>10</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>5</mn> <mn>5</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>45</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>50</mn> <mn>10</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>1221759</mn> </mrow> <mn>10272278170</mn> </mfrac> </mrow> <mo>=</mo> <mn>0.0001189375</mn> <mo>…</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(X=5)=f(5;50,5,10)={{{5 \choose 5}{{45} \choose {5}}} \over {50 \choose 10}}={1\cdot 1221759 \over 10272278170}=0.0001189375\dots ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04901b548401a0de33521577634f83d5ad2bb0be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:74.117ex; height:7.843ex;" alt="{\displaystyle P(X=5)=f(5;50,5,10)={{{5 \choose 5}{{45} \choose {5}}} \over {50 \choose 10}}={1\cdot 1221759 \over 10272278170}=0.0001189375\dots ,}"></span></dd></dl> <p>As expected, the probability of drawing 5 green marbles is roughly 35 times less likely than that of drawing 4. </p> <div class="mw-heading mw-heading3"><h3 id="Symmetries">Symmetries</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hypergeometric_distribution&action=edit&section=6" title="Edit section: Symmetries"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Swapping the roles of green and red marbles: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(k;N,K,n)=f(n-k;N,N-K,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>N</mi> <mo>,</mo> <mi>K</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>;</mo> <mi>N</mi> <mo>,</mo> <mi>N</mi> <mo>−</mo> <mi>K</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(k;N,K,n)=f(n-k;N,N-K,n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93dced621b1351b005b7daa9d628a82a10d0c62e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.088ex; height:2.843ex;" alt="{\displaystyle f(k;N,K,n)=f(n-k;N,N-K,n)}"></span></dd></dl> <p>Swapping the roles of drawn and not drawn marbles: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(k;N,K,n)=f(K-k;N,K,N-n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>N</mi> <mo>,</mo> <mi>K</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo>−</mo> <mi>k</mi> <mo>;</mo> <mi>N</mi> <mo>,</mo> <mi>K</mi> <mo>,</mo> <mi>N</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(k;N,K,n)=f(K-k;N,K,N-n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/329bc4f6dc81aa9f8eeae632b0a192f83a35bf50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.759ex; height:2.843ex;" alt="{\displaystyle f(k;N,K,n)=f(K-k;N,K,N-n)}"></span></dd></dl> <p>Swapping the roles of green and drawn marbles: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(k;N,K,n)=f(k;N,n,K)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>N</mi> <mo>,</mo> <mi>K</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>N</mi> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(k;N,K,n)=f(k;N,n,K)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97c86f413bd424d2e361c2bf12cc88454ad71cbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.949ex; height:2.843ex;" alt="{\displaystyle f(k;N,K,n)=f(k;N,n,K)}"></span></dd></dl> <p>These symmetries generate the <a href="/wiki/Dihedral_group" title="Dihedral group">dihedral group</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 D_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d99bbbdaf59e06536c67afbce7c3f681acd1688" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle D_{4}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Order_of_draws">Order of draws</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hypergeometric_distribution&action=edit&section=7" title="Edit section: Order of draws"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The probability of drawing any set of green and red marbles (the hypergeometric distribution) depends only on the numbers of green and red marbles, not on the order in which they appear; i.e., it is an <a href="/wiki/Exchangeable_random_variables" title="Exchangeable random variables">exchangeable</a> distribution. As a result, the probability of drawing a green marble in the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{\text{th}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>th</mtext> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i^{\text{th}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cc827ec109594f9da6862138d76775cd733866" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.588ex; height:2.676ex;" alt="{\displaystyle i^{\text{th}}}"></span> draw is<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <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(G_{i})={\frac {K}{N}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>K</mi> <mi>N</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(G_{i})={\frac {K}{N}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e865f4877f51b9877e2db83321286f236eebe1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.828ex; height:5.176ex;" alt="{\displaystyle P(G_{i})={\frac {K}{N}}.}"></span></dd></dl> <p>This is an <i>ex ante</i> probability—that is, it is based on not knowing the results of the previous draws. </p> <div class="mw-heading mw-heading3"><h3 id="Tail_bounds">Tail bounds</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hypergeometric_distribution&action=edit&section=8" title="Edit section: Tail bounds"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <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\sim \operatorname {Hypergeometric} (N,K,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mi>Hypergeometric</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo>,</mo> <mi>K</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \operatorname {Hypergeometric} (N,K,n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29261ad981bf9fcf9f689f65ddb8aa5c07421ca5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.508ex; height:2.843ex;" alt="{\displaystyle X\sim \operatorname {Hypergeometric} (N,K,n)}"></span> 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 p=K/N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=K/N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b89ec137bcd3d31f484dfd64800706d779edfd31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:9.649ex; height:2.843ex;" alt="{\displaystyle p=K/N}"></span>. Then for <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 0<t<K/N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>t</mi> <mo><</mo> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<t<K/N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4f72ed17e5fe95a472e93717f8f285f72c70168" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.491ex; height:2.843ex;" alt="{\displaystyle 0<t<K/N}"></span> we can derive the following bounds:<sup id="cite_ref-:0_3-0" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\Pr[X\leq (p-t)n]&\leq e^{-n{\text{D}}(p-t\parallel p)}\leq e^{-2t^{2}n}\\\Pr[X\geq (p+t)n]&\leq e^{-n{\text{D}}(p+t\parallel p)}\leq e^{-2t^{2}n}\\\end{aligned}}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo>≤</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>≤</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>D</mtext> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mi>t</mi> <mo>∥</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>≤</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>n</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo>≥</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>≤</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>D</mtext> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mi>t</mi> <mo>∥</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>≤</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>n</mi> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Pr[X\leq (p-t)n]&\leq e^{-n{\text{D}}(p-t\parallel p)}\leq e^{-2t^{2}n}\\\Pr[X\geq (p+t)n]&\leq e^{-n{\text{D}}(p+t\parallel p)}\leq e^{-2t^{2}n}\\\end{aligned}}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1592b043d16fd6179f3a95d97a3c38806d82cea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:39.773ex; height:7.176ex;" alt="{\displaystyle {\begin{aligned}\Pr[X\leq (p-t)n]&\leq e^{-n{\text{D}}(p-t\parallel p)}\leq e^{-2t^{2}n}\\\Pr[X\geq (p+t)n]&\leq e^{-n{\text{D}}(p+t\parallel p)}\leq e^{-2t^{2}n}\\\end{aligned}}\!}"></span></dd></dl> <p>where </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 D(a\parallel b)=a\log {\frac {a}{b}}+(1-a)\log {\frac {1-a}{1-b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∥</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>a</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>b</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(a\parallel b)=a\log {\frac {a}{b}}+(1-a)\log {\frac {1-a}{1-b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce2e3eb2257394918d8f2a549d4aa582f5040100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:38.251ex; height:5.509ex;" alt="{\displaystyle D(a\parallel b)=a\log {\frac {a}{b}}+(1-a)\log {\frac {1-a}{1-b}}}"></span></dd></dl> <p>is the <a href="/wiki/Kullback-Leibler_divergence" class="mw-redirect" title="Kullback-Leibler divergence">Kullback-Leibler divergence</a> and it is used 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 D(a\parallel b)\geq 2(a-b)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∥</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(a\parallel b)\geq 2(a-b)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53385bd91cc4320297228e400767e8375409cc90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.606ex; height:3.176ex;" alt="{\displaystyle D(a\parallel b)\geq 2(a-b)^{2}}"></span>.<sup id="cite_ref-wordpress.com_4-0" class="reference"><a href="#cite_note-wordpress.com-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p><b>Note</b>: In order to derive the previous bounds, one has to start by observing 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={\frac {\sum _{i=1}^{n}Y_{i}}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X={\frac {\sum _{i=1}^{n}Y_{i}}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd3c84a6e2424353df87095b8d6343fd9a37e7a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.806ex; height:5.843ex;" alt="{\displaystyle X={\frac {\sum _{i=1}^{n}Y_{i}}{n}}}"></span> where <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 Y_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d57be496fff95ee2a97ee43c7f7fe244b4dbf8ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.15ex; height:2.509ex;" alt="{\displaystyle Y_{i}}"></span> are <i>dependent</i> random variables with a specific distribution <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 D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span>. Because most of the theorems about bounds in sum of random variables are concerned with <i>independent</i> sequences of them, one has to first create a sequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b102e069bd5758b0c3d0511c68a12827a36dd761" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.387ex; height:2.509ex;" alt="{\displaystyle Z_{i}}"></span> of <i>independent</i> random variables with the same distribution <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 D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> and apply the theorems on <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'={\frac {\sum _{i=1}^{n}Z_{i}}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X'={\frac {\sum _{i=1}^{n}Z_{i}}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/737f4aca423bb0bb127ecde9d91fdbc111dbcb9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.744ex; height:5.843ex;" alt="{\displaystyle X'={\frac {\sum _{i=1}^{n}Z_{i}}{n}}}"></span>. Then, it is proved from Hoeffding <sup id="cite_ref-:0_3-1" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> that the results and bounds obtained via this process hold for <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> as well. </p><p>If <i>n</i> is larger than <i>N</i>/2, it can be useful to apply symmetry to "invert" the bounds, which give you the following: <sup id="cite_ref-wordpress.com_4-1" class="reference"><a href="#cite_note-wordpress.com-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> <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> </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 {\begin{aligned}\Pr[X\leq (p-t)n]&\leq e^{-(N-n){\text{D}}(p+{\tfrac {tn}{N-n}}||p)}\leq e^{-2t^{2}n{\tfrac {n}{N-n}}}\\\\\Pr[X\geq (p+t)n]&\leq e^{-(N-n){\text{D}}(p-{\tfrac {tn}{N-n}}||p)}\leq e^{-2t^{2}n{\tfrac {n}{N-n}}}\\\end{aligned}}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo>≤</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>≤</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>D</mtext> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>t</mi> <mi>n</mi> </mrow> <mrow> <mi>N</mi> <mo>−</mo> <mi>n</mi> </mrow> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>≤</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>n</mi> <mrow> <mi>N</mi> <mo>−</mo> <mi>n</mi> </mrow> </mfrac> </mstyle> </mrow> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo>≥</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>≤</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>D</mtext> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>t</mi> <mi>n</mi> </mrow> <mrow> <mi>N</mi> <mo>−</mo> <mi>n</mi> </mrow> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>≤</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>n</mi> <mrow> <mi>N</mi> <mo>−</mo> <mi>n</mi> </mrow> </mfrac> </mstyle> </mrow> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Pr[X\leq (p-t)n]&\leq e^{-(N-n){\text{D}}(p+{\tfrac {tn}{N-n}}||p)}\leq e^{-2t^{2}n{\tfrac {n}{N-n}}}\\\\\Pr[X\geq (p+t)n]&\leq e^{-(N-n){\text{D}}(p-{\tfrac {tn}{N-n}}||p)}\leq e^{-2t^{2}n{\tfrac {n}{N-n}}}\\\end{aligned}}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0fd3b1453cac4e9cd1a15a032c154bbb93b7ec4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:52.409ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}\Pr[X\leq (p-t)n]&\leq e^{-(N-n){\text{D}}(p+{\tfrac {tn}{N-n}}||p)}\leq e^{-2t^{2}n{\tfrac {n}{N-n}}}\\\\\Pr[X\geq (p+t)n]&\leq e^{-(N-n){\text{D}}(p-{\tfrac {tn}{N-n}}||p)}\leq e^{-2t^{2}n{\tfrac {n}{N-n}}}\\\end{aligned}}\!}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Statistical_Inference">Statistical Inference</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hypergeometric_distribution&action=edit&section=9" title="Edit section: Statistical Inference"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Hypergeometric_test">Hypergeometric test</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hypergeometric_distribution&action=edit&section=10" title="Edit section: Hypergeometric test"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Fisher%27s_noncentral_hypergeometric_distribution" title="Fisher's noncentral hypergeometric distribution">Fisher's noncentral hypergeometric distribution</a></div> <p>The <b>hypergeometric test</b> uses the hypergeometric distribution to measure the statistical significance of having drawn a sample consisting of a specific number 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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> successes (out 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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> total draws) from a population of size <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> containing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> successes. In a test for over-representation of successes in the sample, the hypergeometric p-value is calculated as the probability of randomly drawing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> or more successes from the population in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> total draws. In a test for under-representation, the p-value is the probability of randomly drawing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> or fewer successes. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Youngronaldfisher2.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Youngronaldfisher2.JPG/200px-Youngronaldfisher2.JPG" decoding="async" width="200" height="280" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/a/aa/Youngronaldfisher2.JPG 1.5x" data-file-width="291" data-file-height="408" /></a><figcaption>Biologist and statistician <a href="/wiki/Ronald_Fisher" title="Ronald Fisher">Ronald Fisher</a></figcaption></figure> <p>The test based on the hypergeometric distribution (hypergeometric test) is identical to the corresponding one-tailed version of <a href="/wiki/Fisher%27s_exact_test" title="Fisher's exact test">Fisher's exact test</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Reciprocally, the p-value of a two-sided Fisher's exact test can be calculated as the sum of two appropriate hypergeometric tests (for more information see<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>). </p><p>The test is often used to identify which sub-populations are over- or under-represented in a sample. This test has a wide range of applications. For example, a marketing group could use the test to understand their customer base by testing a set of known customers for over-representation of various demographic subgroups (e.g., women, people under 30). </p> <div class="mw-heading mw-heading2"><h2 id="Related_distributions">Related distributions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hypergeometric_distribution&action=edit&section=11" title="Edit section: Related distributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <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\sim \operatorname {Hypergeometric} (N,K,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mi>Hypergeometric</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo>,</mo> <mi>K</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \operatorname {Hypergeometric} (N,K,n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29261ad981bf9fcf9f689f65ddb8aa5c07421ca5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.508ex; height:2.843ex;" alt="{\displaystyle X\sim \operatorname {Hypergeometric} (N,K,n)}"></span> 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 p=K/N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=K/N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b89ec137bcd3d31f484dfd64800706d779edfd31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:9.649ex; height:2.843ex;" alt="{\displaystyle p=K/N}"></span>. </p> <ul><li>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 n=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}"></span> then <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> has a <a href="/wiki/Bernoulli_distribution" title="Bernoulli distribution">Bernoulli distribution</a> with parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>.</li> <li>Let <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 Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> have a <a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial distribution</a> with parameters <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> 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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>; this models the number of successes in the analogous sampling problem <i>with</i> replacement. 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 N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> 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 K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> are large compared to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, 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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> is not close to 0 or 1, then <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> 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 Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> have similar distributions, i.e., <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(X\leq k)\approx P(Y\leq k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>≤</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>≤</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(X\leq k)\approx P(Y\leq k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ee6308e1a01f39cd541e79989d956a1bd4ed7f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.581ex; height:2.843ex;" alt="{\displaystyle P(X\leq k)\approx P(Y\leq k)}"></span>.</li> <li>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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> is large, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> 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 K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> are large compared to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, 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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> is not close to 0 or 1, then</li></ul> <dl><dd><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(X\leq k)\approx \Phi \left({\frac {k-np}{\sqrt {np(1-p)}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>≤</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mi mathvariant="normal">Φ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mo>−</mo> <mi>n</mi> <mi>p</mi> </mrow> <msqrt> <mi>n</mi> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(X\leq k)\approx \Phi \left({\frac {k-np}{\sqrt {np(1-p)}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6be8efce5400a7aa643491bf2d8d142ebfd4ac8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.395ex; height:7.509ex;" alt="{\displaystyle P(X\leq k)\approx \Phi \left({\frac {k-np}{\sqrt {np(1-p)}}}\right)}"></span></dd></dl></dd></dl> <p>where <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 \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" 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 \Phi }"></span> is the <a href="/wiki/Standard_normal_distribution#Cumulative_distribution_function" class="mw-redirect" title="Standard normal distribution">standard normal distribution function</a> </p> <ul><li>If the probabilities of drawing a green or red marble are not equal (e.g. because green marbles are bigger/easier to grasp than red marbles) then <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> has a <a href="/wiki/Noncentral_hypergeometric_distribution" class="mw-redirect" title="Noncentral hypergeometric distribution">noncentral hypergeometric distribution</a></li> <li>The <a href="/wiki/Beta-binomial_distribution" title="Beta-binomial distribution">beta-binomial distribution</a> is a <a href="/wiki/Conjugate_prior" title="Conjugate prior">conjugate prior</a> for the hypergeometric distribution.</li></ul> <p>The following table describes four distributions related to the number of successes in a sequence of draws: </p> <table class="wikitable"> <tbody><tr> <th></th> <th>With replacements</th> <th>No replacements </th></tr> <tr> <td>Given number of draws</td> <td><a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial distribution</a></td> <td>hypergeometric distribution </td></tr> <tr> <td>Given number of failures</td> <td><a href="/wiki/Negative_binomial_distribution" title="Negative binomial distribution">negative binomial distribution</a></td> <td><a href="/wiki/Negative_hypergeometric_distribution" title="Negative hypergeometric distribution">negative hypergeometric distribution</a> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Multivariate_hypergeometric_distribution">Multivariate hypergeometric distribution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hypergeometric_distribution&action=edit&section=12" title="Edit section: Multivariate hypergeometric distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1247679731"><table class="infobox infobox-table ib-prob-dist"><caption class="infobox-title">Multivariate hypergeometric distribution</caption><tbody><tr><th scope="row" class="infobox-label"><a href="/wiki/Statistical_parameter" title="Statistical parameter">Parameters</a></th><td colspan="3" class="infobox-data"> <p><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\in \mathbb {N} _{+}=\lbrace 1,2,\ldots \rbrace }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\in \mathbb {N} _{+}=\lbrace 1,2,\ldots \rbrace }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ffdc96f42059da1b2e56f9d08f6bdbdc9be6f3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.576ex; height:2.843ex;" alt="{\displaystyle c\in \mathbb {N} _{+}=\lbrace 1,2,\ldots \rbrace }"></span><br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (K_{1},\ldots ,K_{c})\in \mathbb {N} ^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (K_{1},\ldots ,K_{c})\in \mathbb {N} ^{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5729a15e98e8d92e22433c2dc1f8e86a4027d924" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.395ex; height:2.843ex;" alt="{\displaystyle (K_{1},\ldots ,K_{c})\in \mathbb {N} ^{c}}"></span><br /> </p> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=\sum _{i=1}^{c}K_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </munderover> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=\sum _{i=1}^{c}K_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b88b0e4a78255bbbc61011cd5d0c1d724d815bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:11.677ex; height:6.843ex;" alt="{\displaystyle N=\sum _{i=1}^{c}K_{i}}"></span><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \lbrace 0,\ldots ,N\rbrace }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>N</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \lbrace 0,\ldots ,N\rbrace }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe4d143b89175f81f3e91206297902fb1e7e117f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.965ex; height:2.843ex;" alt="{\displaystyle n\in \lbrace 0,\ldots ,N\rbrace }"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Support_(mathematics)" title="Support (mathematics)">Support</a></th><td colspan="3" class="infobox-data"> <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 \left\{\mathbf {k} \in \left(\mathbb {Z} _{0+}\right)^{c}\,:\,\forall i\ k_{i}\leq K_{i},\sum _{i=1}^{c}k_{i}=n\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>∈</mo> <msup> <mrow> <mo>(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>+</mo> </mrow> </msub> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mo>:</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">∀</mi> <mi>i</mi> <mtext> </mtext> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≤</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </munderover> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>n</mi> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{\mathbf {k} \in \left(\mathbb {Z} _{0+}\right)^{c}\,:\,\forall i\ k_{i}\leq K_{i},\sum _{i=1}^{c}k_{i}=n\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aa33b50c7d6e83dd000157598952b29177e64a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:39.184ex; height:7.509ex;" alt="{\displaystyle \left\{\mathbf {k} \in \left(\mathbb {Z} _{0+}\right)^{c}\,:\,\forall i\ k_{i}\leq K_{i},\sum _{i=1}^{c}k_{i}=n\right\}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Probability_mass_function" title="Probability mass function">PMF</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\prod \limits _{i=1}^{c}{\binom {K_{i}}{k_{i}}}}{\binom {N}{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo movablelimits="false">∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>N</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\prod \limits _{i=1}^{c}{\binom {K_{i}}{k_{i}}}}{\binom {N}{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6f9fd727b32e52d68dd2c4a430e04362eebbcdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:8.041ex; height:10.343ex;" alt="{\displaystyle {\frac {\prod \limits _{i=1}^{c}{\binom {K_{i}}{k_{i}}}}{\binom {N}{n}}}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Expected_value" title="Expected value">Mean</a></th><td colspan="3" class="infobox-data"> <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 \operatorname {E} (k_{i})=n{\frac {K_{i}}{N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>N</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} (k_{i})=n{\frac {K_{i}}{N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e256f0d57d84c6ec55768bf6120ce9d32e4657a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.505ex; height:5.176ex;" alt="{\displaystyle \operatorname {E} (k_{i})=n{\frac {K_{i}}{N}}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Variance" title="Variance">Variance</a></th><td colspan="3" class="infobox-data"> <p><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 \operatorname {Var} (k_{i})=n{\frac {N-n}{N-1}}\;{\frac {K_{i}}{N}}\left(1-{\frac {K_{i}}{N}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>N</mi> <mo>−</mo> <mi>n</mi> </mrow> <mrow> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>N</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>N</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} (k_{i})=n{\frac {N-n}{N-1}}\;{\frac {K_{i}}{N}}\left(1-{\frac {K_{i}}{N}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff60a2779e51543cee080639ad2c1f917eaf70e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.939ex; height:6.176ex;" alt="{\displaystyle \operatorname {Var} (k_{i})=n{\frac {N-n}{N-1}}\;{\frac {K_{i}}{N}}\left(1-{\frac {K_{i}}{N}}\right)}"></span><br /> </p> <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 \operatorname {Cov} (k_{i},k_{j})=-n{\frac {N-n}{N-1}}\;{\frac {K_{i}}{N}}{\frac {K_{j}}{N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>N</mi> <mo>−</mo> <mi>n</mi> </mrow> <mrow> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>N</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>N</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Cov} (k_{i},k_{j})=-n{\frac {N-n}{N-1}}\;{\frac {K_{i}}{N}}{\frac {K_{j}}{N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc27922d3063aeca2320c2a6beb67bc212f9d6b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:32.452ex; height:5.676ex;" alt="{\displaystyle \operatorname {Cov} (k_{i},k_{j})=-n{\frac {N-n}{N-1}}\;{\frac {K_{i}}{N}}{\frac {K_{j}}{N}}}"></span><br /><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 \operatorname {Corr} (k_{i},k_{j})=-{\sqrt {\frac {K_{i}K_{j}}{\left(N-K_{i}\right)\left(N-K_{j}\right)}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Corr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mi>N</mi> <mo>−</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>N</mi> <mo>−</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Corr} (k_{i},k_{j})=-{\sqrt {\frac {K_{i}K_{j}}{\left(N-K_{i}\right)\left(N-K_{j}\right)}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e2da1c00e7a88673bd270715d8b0e07bea2579" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:39.175ex; height:7.676ex;" alt="{\displaystyle \operatorname {Corr} (k_{i},k_{j})=-{\sqrt {\frac {K_{i}K_{j}}{\left(N-K_{i}\right)\left(N-K_{j}\right)}}}}"></span></td></tr></tbody></table> <p>The model of an <a href="/wiki/Urn_problem" title="Urn problem">urn</a> with green and red marbles can be extended to the case where there are more than two colors of marbles. If there are <i>K</i><sub><i>i</i></sub> marbles of color <i>i</i> in the urn and you take <i>n</i> marbles at random without replacement, then the number of marbles of each color in the sample (<i>k</i><sub>1</sub>, <i>k</i><sub>2</sub>,..., <i>k</i><sub><i>c</i></sub>) has the multivariate hypergeometric distribution: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(X_{1}=k_{1},\ldots ,X_{c}=k_{c})={\frac {\prod \limits _{i=1}^{c}{\binom {K_{i}}{k_{i}}}}{\binom {N}{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo movablelimits="false">∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>N</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(X_{1}=k_{1},\ldots ,X_{c}=k_{c})={\frac {\prod \limits _{i=1}^{c}{\binom {K_{i}}{k_{i}}}}{\binom {N}{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83fa370f0c2d8e76b1e1b914881aa089681d5986" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:37.086ex; height:10.343ex;" alt="{\displaystyle \Pr(X_{1}=k_{1},\ldots ,X_{c}=k_{c})={\frac {\prod \limits _{i=1}^{c}{\binom {K_{i}}{k_{i}}}}{\binom {N}{n}}}}"></span></dd></dl> <p>This has the same relationship to the <a href="/wiki/Multinomial_distribution" title="Multinomial distribution">multinomial distribution</a> that the hypergeometric distribution has to the binomial distribution—the multinomial distribution is the "with-replacement" distribution and the multivariate hypergeometric is the "without-replacement" distribution. </p><p>The properties of this distribution are given in the adjacent table,<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> where <i>c</i> is the number of different colors 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 N=\sum _{i=1}^{c}K_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </munderover> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=\sum _{i=1}^{c}K_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b88b0e4a78255bbbc61011cd5d0c1d724d815bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:11.677ex; height:6.843ex;" alt="{\displaystyle N=\sum _{i=1}^{c}K_{i}}"></span> is the total number of marbles in the urn. </p> <div class="mw-heading mw-heading4"><h4 id="Example">Example</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hypergeometric_distribution&action=edit&section=13" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose there are 5 black, 10 white, and 15 red marbles in an urn. If six marbles are chosen without replacement, the probability that exactly two of each color are chosen is </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(2{\text{ black}},2{\text{ white}},2{\text{ red}})={{{5 \choose 2}{10 \choose 2}{15 \choose 2}} \over {30 \choose 6}}=0.079575596816976}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> black</mtext> </mrow> <mo>,</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> white</mtext> </mrow> <mo>,</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> red</mtext> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>5</mn> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>10</mn> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>15</mn> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>30</mn> <mn>6</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0.079575596816976</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(2{\text{ black}},2{\text{ white}},2{\text{ red}})={{{5 \choose 2}{10 \choose 2}{15 \choose 2}} \over {30 \choose 6}}=0.079575596816976}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1c732a6291d8d06d36d382de45c024b1b769a4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:61.783ex; height:7.676ex;" alt="{\displaystyle P(2{\text{ black}},2{\text{ white}},2{\text{ red}})={{{5 \choose 2}{10 \choose 2}{15 \choose 2}} \over {30 \choose 6}}=0.079575596816976}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Occurrence_and_applications">Occurrence and applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hypergeometric_distribution&action=edit&section=14" title="Edit section: Occurrence and applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Application_to_auditing_elections">Application to auditing elections</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hypergeometric_distribution&action=edit&section=15" title="Edit section: Application to auditing elections"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Election_Samples.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Election_Samples.png/220px-Election_Samples.png" decoding="async" width="220" height="158" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Election_Samples.png/330px-Election_Samples.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/95/Election_Samples.png/440px-Election_Samples.png 2x" data-file-width="1848" data-file-height="1328" /></a><figcaption>Samples used for election audits and resulting chance of missing a problem</figcaption></figure> <p><a href="/wiki/Election_audits" class="mw-redirect" title="Election audits">Election audits</a> typically test a sample of machine-counted precincts to see if recounts by hand or machine match the original counts. Mismatches result in either a report or a larger recount. The sampling rates are usually defined by law, not statistical design, so for a legally defined sample size <span class="texhtml mvar" style="font-style:italic;">n</span>, what is the probability of missing a problem which is present in <span class="texhtml mvar" style="font-style:italic;">K</span> precincts, such as a hack or bug? This is the probability that <span class="texhtml"><i>k</i> = 0 .</span> Bugs are often obscure, and a hacker can minimize detection by affecting only a few precincts, which will still affect close elections, so a plausible scenario is for <span class="texhtml mvar" style="font-style:italic;">K</span> to be on the order of 5% of <span class="texhtml mvar" style="font-style:italic;">N</span>. Audits typically cover 1% to 10% of precincts (often 3%),<sup id="cite_ref-newyorkaudit_9-0" class="reference"><a href="#cite_note-newyorkaudit-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-vvstates_10-0" class="reference"><a href="#cite_note-vvstates-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ncsl_11-0" class="reference"><a href="#cite_note-ncsl-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> so they have a high chance of missing a problem. For example, if a problem is present in 5 of 100 precincts, a 3% sample has 86% probability that <span class="nowrap"> <span class="texhtml"> <i>k</i> = 0 </span> </span> so the problem would not be noticed, and only 14% probability of the problem appearing in the sample (positive <span class="texhtml mvar" style="font-style:italic;"> k </span>): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {\boldsymbol {\mathcal {P}}} \{\ X=0\ \}&={\frac {\ \left[\ {\binom {\text{Hack}}{0}}{\binom {N\ -\ {\text{Hack}}}{n\ -\ 0}}\ \right]\ }{\left[\ {\binom {N}{n}}\ \right]}}={\frac {\ \left[\ {\binom {N\ -\ {\text{Hack}}}{n}}\ \right]}{\ \left[\ {\binom {N}{n}}\ \right]\ }}={\frac {\ \left[\ {\frac {\ (N\ -\ {\text{Hack}})!\ }{n!(N\ -\ {\text{Hack}}-n)!}}\ \right]\ }{\left[\ {\frac {N!}{n!(N\ -\ n)!}}\ \right]}}={\frac {\ \left[\ {\frac {(N-{\text{Hack}})!}{(N\ -\ {\text{Hack}}\ -\ n)!}}\ \right]\ }{\left[\ {\frac {N!}{(N\ -\ n)!}}\ \right]}}\\[8pt]&={\frac {\ \left[\ {\binom {100-5}{3}}\ \right]\ }{\ \left[\ {\binom {100}{3}}\ \right]\ }}={\frac {\ \left[\ {\frac {(100-5)!}{(100-5-3)!}}\ \right]\ }{\left[\ {\frac {100!}{(100-3)!}}\ \right]}}={\frac {\ \left[\ {\frac {95!}{92!}}\ \right]\ }{\ \left[\ {\frac {100!}{97!}}\ \right]\ }}={\frac {\ 95\times 94\times 93\ }{100\times 99\times 98}}=86\%\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.1em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic-bold" mathvariant="bold-script">P</mi> </mrow> </mrow> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mtext> </mtext> <mi>X</mi> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mo fence="false" stretchy="false">}</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mrow> <mo>[</mo> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mtext>Hack</mtext> <mn>0</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>N</mi> <mtext> </mtext> <mo>−</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>Hack</mtext> </mrow> </mrow> <mrow> <mi>n</mi> <mtext> </mtext> <mo>−</mo> <mtext> </mtext> <mn>0</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mtext> </mtext> </mrow> <mo>]</mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mo>[</mo> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>N</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mtext> </mtext> </mrow> <mo>]</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mrow> <mo>[</mo> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>N</mi> <mtext> </mtext> <mo>−</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>Hack</mtext> </mrow> </mrow> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mtext> </mtext> </mrow> <mo>]</mo> </mrow> </mrow> <mrow> <mtext> </mtext> <mrow> <mo>[</mo> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>N</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mtext> </mtext> </mrow> <mo>]</mo> </mrow> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mrow> <mo>[</mo> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>N</mi> <mtext> </mtext> <mo>−</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>Hack</mtext> </mrow> <mo stretchy="false">)</mo> <mo>!</mo> <mtext> </mtext> </mrow> <mrow> <mi>n</mi> <mo>!</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mtext> </mtext> <mo>−</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>Hack</mtext> </mrow> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mtext> </mtext> </mrow> <mo>]</mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mo>[</mo> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>N</mi> <mo>!</mo> </mrow> <mrow> <mi>n</mi> <mo>!</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mtext> </mtext> <mo>−</mo> <mtext> </mtext> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mtext> </mtext> </mrow> <mo>]</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mrow> <mo>[</mo> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Hack</mtext> </mrow> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mtext> </mtext> <mo>−</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>Hack</mtext> </mrow> <mtext> </mtext> <mo>−</mo> <mtext> </mtext> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mtext> </mtext> </mrow> <mo>]</mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mo>[</mo> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>N</mi> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mtext> </mtext> <mo>−</mo> <mtext> </mtext> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mtext> </mtext> </mrow> <mo>]</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mrow> <mo>[</mo> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>100</mn> <mo>−</mo> <mn>5</mn> </mrow> <mn>3</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mtext> </mtext> </mrow> <mo>]</mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mtext> </mtext> <mrow> <mo>[</mo> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>100</mn> <mn>3</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mtext> </mtext> </mrow> <mo>]</mo> </mrow> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mrow> <mo>[</mo> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>100</mn> <mo>−</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>100</mn> <mo>−</mo> <mn>5</mn> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mtext> </mtext> </mrow> <mo>]</mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mo>[</mo> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>100</mn> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>100</mn> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mtext> </mtext> </mrow> <mo>]</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mrow> <mo>[</mo> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>95</mn> <mo>!</mo> </mrow> <mrow> <mn>92</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mtext> </mtext> </mrow> <mo>]</mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mtext> </mtext> <mrow> <mo>[</mo> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>100</mn> <mo>!</mo> </mrow> <mrow> <mn>97</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mtext> </mtext> </mrow> <mo>]</mo> </mrow> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mn>95</mn> <mo>×</mo> <mn>94</mn> <mo>×</mo> <mn>93</mn> <mtext> </mtext> </mrow> <mrow> <mn>100</mn> <mo>×</mo> <mn>99</mn> <mo>×</mo> <mn>98</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>86</mn> <mi mathvariant="normal">%</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {\boldsymbol {\mathcal {P}}} \{\ X=0\ \}&={\frac {\ \left[\ {\binom {\text{Hack}}{0}}{\binom {N\ -\ {\text{Hack}}}{n\ -\ 0}}\ \right]\ }{\left[\ {\binom {N}{n}}\ \right]}}={\frac {\ \left[\ {\binom {N\ -\ {\text{Hack}}}{n}}\ \right]}{\ \left[\ {\binom {N}{n}}\ \right]\ }}={\frac {\ \left[\ {\frac {\ (N\ -\ {\text{Hack}})!\ }{n!(N\ -\ {\text{Hack}}-n)!}}\ \right]\ }{\left[\ {\frac {N!}{n!(N\ -\ n)!}}\ \right]}}={\frac {\ \left[\ {\frac {(N-{\text{Hack}})!}{(N\ -\ {\text{Hack}}\ -\ n)!}}\ \right]\ }{\left[\ {\frac {N!}{(N\ -\ n)!}}\ \right]}}\\[8pt]&={\frac {\ \left[\ {\binom {100-5}{3}}\ \right]\ }{\ \left[\ {\binom {100}{3}}\ \right]\ }}={\frac {\ \left[\ {\frac {(100-5)!}{(100-5-3)!}}\ \right]\ }{\left[\ {\frac {100!}{(100-3)!}}\ \right]}}={\frac {\ \left[\ {\frac {95!}{92!}}\ \right]\ }{\ \left[\ {\frac {100!}{97!}}\ \right]\ }}={\frac {\ 95\times 94\times 93\ }{100\times 99\times 98}}=86\%\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9b3abff1a0a9e4352226df70d8310724cb4a33c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.505ex; width:100.91ex; height:22.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {\boldsymbol {\mathcal {P}}} \{\ X=0\ \}&={\frac {\ \left[\ {\binom {\text{Hack}}{0}}{\binom {N\ -\ {\text{Hack}}}{n\ -\ 0}}\ \right]\ }{\left[\ {\binom {N}{n}}\ \right]}}={\frac {\ \left[\ {\binom {N\ -\ {\text{Hack}}}{n}}\ \right]}{\ \left[\ {\binom {N}{n}}\ \right]\ }}={\frac {\ \left[\ {\frac {\ (N\ -\ {\text{Hack}})!\ }{n!(N\ -\ {\text{Hack}}-n)!}}\ \right]\ }{\left[\ {\frac {N!}{n!(N\ -\ n)!}}\ \right]}}={\frac {\ \left[\ {\frac {(N-{\text{Hack}})!}{(N\ -\ {\text{Hack}}\ -\ n)!}}\ \right]\ }{\left[\ {\frac {N!}{(N\ -\ n)!}}\ \right]}}\\[8pt]&={\frac {\ \left[\ {\binom {100-5}{3}}\ \right]\ }{\ \left[\ {\binom {100}{3}}\ \right]\ }}={\frac {\ \left[\ {\frac {(100-5)!}{(100-5-3)!}}\ \right]\ }{\left[\ {\frac {100!}{(100-3)!}}\ \right]}}={\frac {\ \left[\ {\frac {95!}{92!}}\ \right]\ }{\ \left[\ {\frac {100!}{97!}}\ \right]\ }}={\frac {\ 95\times 94\times 93\ }{100\times 99\times 98}}=86\%\end{aligned}}}"></span></dd></dl> <p>The sample would need 45 precincts in order to have probability under 5% that <i>k</i> = 0 in the sample, and thus have probability over 95% of finding the problem: </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 \operatorname {\boldsymbol {\mathcal {P}}} \{\ X=0\ \}={\frac {\ \left[\ {\binom {100-5}{45}}\ \right]\ }{\left[\ {\binom {100}{45}}\ \right]}}={\frac {\ \left[\ {\frac {95!}{50!}}\ \right]\ }{\left[\ {\frac {100!}{55!}}\ \right]}}={\frac {\ 95\times 94\times \cdots \times 51\ }{\ 100\times 99\times \cdots \times 56\ }}={\frac {\ 55\times 54\times 53\times 52\times 51\ }{\ 100\times 99\times 98\times 97\times 96\ }}=4.6\%~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic-bold" mathvariant="bold-script">P</mi> </mrow> </mrow> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mtext> </mtext> <mi>X</mi> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mrow> <mo>[</mo> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>100</mn> <mo>−</mo> <mn>5</mn> </mrow> <mn>45</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mtext> </mtext> </mrow> <mo>]</mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mo>[</mo> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>100</mn> <mn>45</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mtext> </mtext> </mrow> <mo>]</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mrow> <mo>[</mo> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>95</mn> <mo>!</mo> </mrow> <mrow> <mn>50</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mtext> </mtext> </mrow> <mo>]</mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mo>[</mo> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>100</mn> <mo>!</mo> </mrow> <mrow> <mn>55</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mtext> </mtext> </mrow> <mo>]</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mn>95</mn> <mo>×</mo> <mn>94</mn> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <mn>51</mn> <mtext> </mtext> </mrow> <mrow> <mtext> </mtext> <mn>100</mn> <mo>×</mo> <mn>99</mn> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <mn>56</mn> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mn>55</mn> <mo>×</mo> <mn>54</mn> <mo>×</mo> <mn>53</mn> <mo>×</mo> <mn>52</mn> <mo>×</mo> <mn>51</mn> <mtext> </mtext> </mrow> <mrow> <mtext> </mtext> <mn>100</mn> <mo>×</mo> <mn>99</mn> <mo>×</mo> <mn>98</mn> <mo>×</mo> <mn>97</mn> <mo>×</mo> <mn>96</mn> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>4.6</mn> <mi mathvariant="normal">%</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {\boldsymbol {\mathcal {P}}} \{\ X=0\ \}={\frac {\ \left[\ {\binom {100-5}{45}}\ \right]\ }{\left[\ {\binom {100}{45}}\ \right]}}={\frac {\ \left[\ {\frac {95!}{50!}}\ \right]\ }{\left[\ {\frac {100!}{55!}}\ \right]}}={\frac {\ 95\times 94\times \cdots \times 51\ }{\ 100\times 99\times \cdots \times 56\ }}={\frac {\ 55\times 54\times 53\times 52\times 51\ }{\ 100\times 99\times 98\times 97\times 96\ }}=4.6\%~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/544dcc2ed3b484ae01c52ceee81677d51480999b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:102.48ex; height:10.176ex;" alt="{\displaystyle \operatorname {\boldsymbol {\mathcal {P}}} \{\ X=0\ \}={\frac {\ \left[\ {\binom {100-5}{45}}\ \right]\ }{\left[\ {\binom {100}{45}}\ \right]}}={\frac {\ \left[\ {\frac {95!}{50!}}\ \right]\ }{\left[\ {\frac {100!}{55!}}\ \right]}}={\frac {\ 95\times 94\times \cdots \times 51\ }{\ 100\times 99\times \cdots \times 56\ }}={\frac {\ 55\times 54\times 53\times 52\times 51\ }{\ 100\times 99\times 98\times 97\times 96\ }}=4.6\%~.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Application_to_Texas_hold'em_poker"><span id="Application_to_Texas_hold.27em_poker"></span>Application to Texas hold'em poker</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hypergeometric_distribution&action=edit&section=16" title="Edit section: Application to Texas hold'em poker"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Hold%27em" class="mw-redirect" title="Hold'em">hold'em</a> poker players make the best hand they can combining the two cards in their hand with the 5 cards (community cards) eventually turned up on the table. The deck has 52 and there are 13 of each suit. For this example assume a player has 2 clubs in the hand and there are 3 cards showing on the table, 2 of which are also clubs. The player would like to know the probability of one of the next 2 cards to be shown being a club to complete the <a href="/wiki/Flush_(poker)" class="mw-redirect" title="Flush (poker)">flush</a>.<br /> (Note that the probability calculated in this example assumes no information is known about the cards in the other players' hands; however, experienced poker players may consider how the other players place their bets (check, call, raise, or fold) in considering the probability for each scenario. Strictly speaking, the approach to calculating success probabilities outlined here is accurate in a scenario where there is just one player at the table; in a multiplayer game this probability might be adjusted somewhat based on the betting play of the opponents.) </p><p>There are 4 clubs showing so there are 9 clubs still unseen. There are 5 cards showing (2 in the hand and 3 on the table) so there are <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 52-5=47}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>52</mn> <mo>−</mo> <mn>5</mn> <mo>=</mo> <mn>47</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 52-5=47}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb3e7717ee3a9a320853fcba194466f49314fe0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.751ex; height:2.343ex;" alt="{\displaystyle 52-5=47}"></span> still unseen. </p><p>The probability that one of the next two cards turned is a club can be calculated using hypergeometric with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1,n=2,K=9}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>K</mi> <mo>=</mo> <mn>9</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1,n=2,K=9}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02602a0149937c132d4b366aa2c0b6ac11e541b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.523ex; height:2.509ex;" alt="{\displaystyle k=1,n=2,K=9}"></span> 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 N=47}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mn>47</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=47}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b39eb879f360293549f89b027fee61b2b555989" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.487ex; height:2.176ex;" alt="{\displaystyle N=47}"></span>. (about 31.64%) </p><p>The probability that both of the next two cards turned are clubs can be calculated using hypergeometric with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=2,n=2,K=9}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>K</mi> <mo>=</mo> <mn>9</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=2,n=2,K=9}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0937629a5bd7f76a749f0f0d7ff72dee5195239" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.523ex; height:2.509ex;" alt="{\displaystyle k=2,n=2,K=9}"></span> 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 N=47}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mn>47</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=47}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b39eb879f360293549f89b027fee61b2b555989" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.487ex; height:2.176ex;" alt="{\displaystyle N=47}"></span>. (about 3.33%) </p><p>The probability that neither of the next two cards turned are clubs can be calculated using hypergeometric with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=0,n=2,K=9}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>K</mi> <mo>=</mo> <mn>9</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=0,n=2,K=9}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ba2fa49e35d229ca8c3bdc8e30ce2472f8d92be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.523ex; height:2.509ex;" alt="{\displaystyle k=0,n=2,K=9}"></span> 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 N=47}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mn>47</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=47}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b39eb879f360293549f89b027fee61b2b555989" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.487ex; height:2.176ex;" alt="{\displaystyle N=47}"></span>. (about 65.03%) </p> <div class="mw-heading mw-heading3"><h3 id="Application_to_Keno">Application to Keno</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hypergeometric_distribution&action=edit&section=17" title="Edit section: Application to Keno"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The hypergeometric distribution is indispensable for calculating <a href="/wiki/Keno" title="Keno">Keno</a> odds. In Keno, 20 balls are randomly drawn from a collection of 80 numbered balls in a container, rather like <a href="/wiki/Bingo_(American_version)" title="Bingo (American version)">American Bingo</a>. Prior to each draw, a player selects a certain number of <i>spots</i> by marking a paper form supplied for this purpose. For example, a player might <i>play a 6-spot</i> by marking 6 numbers, each from a range of 1 through 80 inclusive. Then (after all players have taken their forms to a cashier and been given a duplicate of their marked form, and paid their wager) 20 balls are drawn. Some of the balls drawn may match some or all of the balls selected by the player. Generally speaking, the more <i>hits</i> (balls drawn that match player numbers selected) the greater the payoff. </p><p>For example, if a customer bets ("plays") $1 for a 6-spot (not an uncommon example) and hits 4 out of the 6, the casino would pay out $4. Payouts can vary from one casino to the next, but $4 is a typical value here. The probability of this event is: </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(X=4)=f(4;80,6,20)={{{6 \choose 4}{{80-6} \choose {20-4}}} \over {80 \choose 20}}\approx 0.02853791}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>=</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo>;</mo> <mn>80</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>20</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>6</mn> <mn>4</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>80</mn> <mo>−</mo> <mn>6</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>20</mn> <mo>−</mo> <mn>4</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>80</mn> <mn>20</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>≈</mo> <mn>0.02853791</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(X=4)=f(4;80,6,20)={{{6 \choose 4}{{80-6} \choose {20-4}}} \over {80 \choose 20}}\approx 0.02853791}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0cfb913fd46f904a6aa5d53b1909c3c44a0e13e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:53.027ex; height:7.843ex;" alt="{\displaystyle P(X=4)=f(4;80,6,20)={{{6 \choose 4}{{80-6} \choose {20-4}}} \over {80 \choose 20}}\approx 0.02853791}"></span></dd></dl> <p>Similarly, the chance for hitting 5 spots out of 6 selected is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {{{6 \choose 5}{{74} \choose {15}}} \over {80 \choose 20}}\approx 0.003095639}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>6</mn> <mn>5</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>74</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>80</mn> <mn>20</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>≈</mo> <mn>0.003095639</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {{{6 \choose 5}{{74} \choose {15}}} \over {80 \choose 20}}\approx 0.003095639}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbee7b02cd410700ecd4dd551c1f396c702758fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:22.932ex; height:7.843ex;" alt="{\displaystyle {{{6 \choose 5}{{74} \choose {15}}} \over {80 \choose 20}}\approx 0.003095639}"></span> while a typical payout might be $88. The payout for hitting all 6 would be around $1500 (probability ≈ 0.000128985 or 7752-to-1). The only other nonzero payout might be $1 for hitting 3 numbers (i.e., you get your bet back), which has a probability near 0.129819548. </p><p>Taking the sum of products of payouts times corresponding probabilities we get an expected return of 0.70986492 or roughly 71% for a 6-spot, for a house advantage of 29%. Other spots-played have a similar expected return. This very poor return (for the player) is usually explained by the large overhead (floor space, equipment, personnel) required for the game. </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=Hypergeometric_distribution&action=edit&section=18" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Noncentral_hypergeometric_distributions" title="Noncentral hypergeometric distributions">Noncentral hypergeometric distributions</a></li> <li><a href="/wiki/Negative_hypergeometric_distribution" title="Negative hypergeometric distribution">Negative hypergeometric distribution</a></li> <li><a href="/wiki/Multinomial_distribution" title="Multinomial distribution">Multinomial distribution</a></li> <li><a href="/wiki/Sampling_(statistics)" title="Sampling (statistics)">Sampling (statistics)</a></li> <li><a href="/wiki/Generalized_hypergeometric_function" title="Generalized hypergeometric function">Generalized hypergeometric function</a></li> <li><a href="/wiki/Coupon_collector%27s_problem" title="Coupon collector's problem">Coupon collector's problem</a></li> <li><a href="/wiki/Geometric_distribution" title="Geometric distribution">Geometric distribution</a></li> <li><a href="/wiki/Keno" title="Keno">Keno</a></li> <li><a href="/wiki/Lady_tasting_tea" title="Lady tasting tea">Lady tasting tea</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hypergeometric_distribution&action=edit&section=19" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hypergeometric_distribution&action=edit&section=20" title="Edit section: Citations"><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 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Elsevier. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2139%2Fssrn.3536011">10.2139/ssrn.3536011</a>. <a href="/wiki/SSRN_(identifier)" class="mw-redirect" title="SSRN (identifier)">SSRN</a> <a rel="nofollow" class="external text" href="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3536011">3536011</a>. SSRN 3536011<span class="reference-accessdate">. Retrieved <span class="nowrap">4 December</span> 2023</span> – via SSRN.com.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=report&rft.btitle=Start+spreading+the+news%3A+New+York%27s+post-election+audit+has+major+flaws&rft.pub=Elsevier&rft.date=2020-02-10&rft_id=https%3A%2F%2Fpapers.ssrn.com%2Fsol3%2Fpapers.cfm%3Fabstract_id%3D3536011%23id-name%3DSSRN&rft_id=info%3Adoi%2F10.2139%2Fssrn.3536011&rft.aulast=Glazer&rft.aufirst=Amanda&rft.au=Spertus%2C+Jacob&rft_id=https%3A%2F%2Fpapers.ssrn.com%2Fsol3%2Fpapers.cfm%3Fabstract_id%3D3536011&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHypergeometric+distribution" class="Z3988"></span></span> </li> <li id="cite_note-vvstates-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-vvstates_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20200104201852/https://www.verifiedvoting.org/state-audit-laws/">"State audit laws"</a>. <i>Verified Voting</i>. 10 February 2017. 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Retrieved <span class="nowrap">2 April</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Verified+Voting&rft.atitle=State+audit+laws&rft.date=2017-02-10&rft_id=https%3A%2F%2Fwww.verifiedvoting.org%2Fstate-audit-laws%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHypergeometric+distribution" class="Z3988"></span></span> </li> <li id="cite_note-ncsl-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-ncsl_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.ncsl.org/research/elections-and-campaigns/post-election-audits635926066.aspx#state">"Post-election audits"</a>. <i>ncsl.org</i>. National Conference of State Legislatures<span class="reference-accessdate">. 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class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hypergeometric_distribution&action=edit&section=21" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerkopec2007" class="citation journal cs1">Berkopec, Aleš (2007). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jda.2006.01.001">"HyperQuick algorithm for discrete hypergeometric distribution"</a>. <i>Journal of Discrete Algorithms</i>. <b>5</b> (2): 341–347. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jda.2006.01.001">10.1016/j.jda.2006.01.001</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Discrete+Algorithms&rft.atitle=HyperQuick+algorithm+for+discrete+hypergeometric+distribution&rft.volume=5&rft.issue=2&rft.pages=341-347&rft.date=2007&rft_id=info%3Adoi%2F10.1016%2Fj.jda.2006.01.001&rft.aulast=Berkopec&rft.aufirst=Ale%C5%A1&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252Fj.jda.2006.01.001&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHypergeometric+distribution" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSkala2011" class="citation arxiv cs1">Skala, M. (2011). "Hypergeometric tail inequalities: ending the insanity". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1311.5939">1311.5939</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.PR">math.PR</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Hypergeometric+tail+inequalities%3A+ending+the+insanity&rft.date=2011&rft_id=info%3Aarxiv%2F1311.5939&rft.aulast=Skala&rft.aufirst=M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHypergeometric+distribution" class="Z3988"></span> unpublished note</li></ul> </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=Hypergeometric_distribution&action=edit&section=22" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/TheHypergeometricDistribution/">The Hypergeometric Distribution</a> and <a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/BinomialApproximationToAHypergeometricRandomVariable/">Binomial Approximation to a Hypergeometric Random Variable</a> by Chris Boucher, <a href="/wiki/Wolfram_Demonstrations_Project" title="Wolfram Demonstrations Project">Wolfram Demonstrations Project</a>.</li> <li><span class="citation mathworld" id="Reference-Mathworld-Hypergeometric_Distribution"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/HypergeometricDistribution.html">"Hypergeometric Distribution"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Hypergeometric+Distribution&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FHypergeometricDistribution.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHypergeometric+distribution" class="Z3988"></span></span></li></ul> <div style="clear:both;" class=""></div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output 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distributions"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Probability_distributions" title="Special:EditPage/Template:Probability distributions"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Probability_distributions_(list)" style="font-size:114%;margin:0 4em"><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distributions</a> (<a href="/wiki/List_of_probability_distributions" title="List of probability distributions">list</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Discrete <br />univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">with finite <br />support</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Benford%27s_law" title="Benford's law">Benford</a></li> <li><a href="/wiki/Bernoulli_distribution" title="Bernoulli distribution">Bernoulli</a></li> <li><a href="/wiki/Beta-binomial_distribution" title="Beta-binomial distribution">Beta-binomial</a></li> <li><a href="/wiki/Binomial_distribution" title="Binomial distribution">Binomial</a></li> <li><a href="/wiki/Categorical_distribution" title="Categorical distribution">Categorical</a></li> <li><a class="mw-selflink selflink">Hypergeometric</a> <ul><li><a href="/wiki/Negative_hypergeometric_distribution" title="Negative hypergeometric distribution">Negative</a></li></ul></li> <li><a href="/wiki/Poisson_binomial_distribution" title="Poisson binomial distribution">Poisson binomial</a></li> <li><a href="/wiki/Rademacher_distribution" title="Rademacher distribution">Rademacher</a></li> <li><a href="/wiki/Soliton_distribution" title="Soliton distribution">Soliton</a></li> <li><a href="/wiki/Discrete_uniform_distribution" title="Discrete uniform distribution">Discrete uniform</a></li> <li><a href="/wiki/Zipf%27s_law" title="Zipf's law">Zipf</a></li> <li><a href="/wiki/Zipf%E2%80%93Mandelbrot_law" title="Zipf–Mandelbrot law">Zipf–Mandelbrot</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">with infinite <br />support</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Beta_negative_binomial_distribution" title="Beta negative binomial distribution">Beta negative binomial</a></li> <li><a href="/wiki/Borel_distribution" title="Borel distribution">Borel</a></li> <li><a href="/wiki/Conway%E2%80%93Maxwell%E2%80%93Poisson_distribution" title="Conway–Maxwell–Poisson distribution">Conway–Maxwell–Poisson</a></li> <li><a href="/wiki/Discrete_phase-type_distribution" title="Discrete phase-type distribution">Discrete phase-type</a></li> <li><a href="/wiki/Delaporte_distribution" title="Delaporte distribution">Delaporte</a></li> <li><a href="/wiki/Extended_negative_binomial_distribution" title="Extended negative binomial distribution">Extended negative binomial</a></li> <li><a href="/wiki/Flory%E2%80%93Schulz_distribution" title="Flory–Schulz distribution">Flory–Schulz</a></li> <li><a href="/wiki/Gauss%E2%80%93Kuzmin_distribution" title="Gauss–Kuzmin distribution">Gauss–Kuzmin</a></li> <li><a href="/wiki/Geometric_distribution" title="Geometric distribution">Geometric</a></li> <li><a href="/wiki/Logarithmic_distribution" title="Logarithmic distribution">Logarithmic</a></li> <li><a href="/wiki/Mixed_Poisson_distribution" title="Mixed Poisson distribution">Mixed Poisson</a></li> <li><a href="/wiki/Negative_binomial_distribution" title="Negative binomial distribution">Negative binomial</a></li> <li><a href="/wiki/(a,b,0)_class_of_distributions" title="(a,b,0) class of distributions">Panjer</a></li> <li><a href="/wiki/Parabolic_fractal_distribution" title="Parabolic fractal distribution">Parabolic fractal</a></li> <li><a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson</a></li> <li><a href="/wiki/Skellam_distribution" title="Skellam distribution">Skellam</a></li> <li><a href="/wiki/Yule%E2%80%93Simon_distribution" title="Yule–Simon distribution">Yule–Simon</a></li> <li><a href="/wiki/Zeta_distribution" title="Zeta distribution">Zeta</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Continuous <br />univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">supported on a <br />bounded interval</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arcsine_distribution" title="Arcsine distribution">Arcsine</a></li> <li><a href="/wiki/ARGUS_distribution" title="ARGUS distribution">ARGUS</a></li> <li><a href="/wiki/Balding%E2%80%93Nichols_model" title="Balding–Nichols model">Balding–Nichols</a></li> <li><a href="/wiki/Bates_distribution" title="Bates distribution">Bates</a></li> <li><a href="/wiki/Beta_distribution" title="Beta distribution">Beta</a> <ul><li><a href="/wiki/Generalized_beta_distribution" title="Generalized beta distribution">Generalized</a></li></ul></li> <li><a href="/wiki/Beta_rectangular_distribution" title="Beta rectangular distribution">Beta rectangular</a></li> <li><a href="/wiki/Continuous_Bernoulli_distribution" title="Continuous Bernoulli distribution">Continuous Bernoulli</a></li> <li><a href="/wiki/Irwin%E2%80%93Hall_distribution" title="Irwin–Hall distribution">Irwin–Hall</a></li> <li><a href="/wiki/Kumaraswamy_distribution" title="Kumaraswamy distribution">Kumaraswamy</a></li> <li><a href="/wiki/Logit-normal_distribution" title="Logit-normal distribution">Logit-normal</a></li> <li><a href="/wiki/Noncentral_beta_distribution" title="Noncentral beta distribution">Noncentral beta</a></li> <li><a href="/wiki/PERT_distribution" title="PERT distribution">PERT</a></li> <li><a href="/wiki/Raised_cosine_distribution" title="Raised cosine distribution">Raised cosine</a></li> <li><a href="/wiki/Reciprocal_distribution" title="Reciprocal distribution">Reciprocal</a></li> <li><a href="/wiki/Triangular_distribution" title="Triangular distribution">Triangular</a></li> <li><a href="/wiki/U-quadratic_distribution" title="U-quadratic distribution">U-quadratic</a></li> <li><a href="/wiki/Continuous_uniform_distribution" title="Continuous uniform distribution">Uniform</a></li> <li><a href="/wiki/Wigner_semicircle_distribution" title="Wigner semicircle distribution">Wigner semicircle</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">supported on a <br />semi-infinite <br />interval</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Benini_distribution" title="Benini distribution">Benini</a></li> <li><a href="/wiki/Benktander_type_I_distribution" title="Benktander type I distribution">Benktander 1st kind</a></li> <li><a href="/wiki/Benktander_type_II_distribution" title="Benktander type II distribution">Benktander 2nd kind</a></li> <li><a href="/wiki/Beta_prime_distribution" title="Beta prime distribution">Beta prime</a></li> <li><a href="/wiki/Burr_distribution" title="Burr distribution">Burr</a></li> <li><a href="/wiki/Chi_distribution" title="Chi distribution">Chi</a></li> <li><a href="/wiki/Chi-squared_distribution" title="Chi-squared distribution">Chi-squared</a> <ul><li><a href="/wiki/Noncentral_chi-squared_distribution" title="Noncentral chi-squared distribution">Noncentral</a></li> <li><a href="/wiki/Inverse-chi-squared_distribution" title="Inverse-chi-squared distribution">Inverse</a> <ul><li><a href="/wiki/Scaled_inverse_chi-squared_distribution" title="Scaled inverse chi-squared distribution">Scaled</a></li></ul></li></ul></li> <li><a href="/wiki/Dagum_distribution" title="Dagum distribution">Dagum</a></li> <li><a href="/wiki/Davis_distribution" title="Davis distribution">Davis</a></li> <li><a href="/wiki/Erlang_distribution" title="Erlang distribution">Erlang</a> <ul><li><a href="/wiki/Hyper-Erlang_distribution" title="Hyper-Erlang distribution">Hyper</a></li></ul></li> <li><a href="/wiki/Exponential_distribution" title="Exponential distribution">Exponential</a> <ul><li><a href="/wiki/Hyperexponential_distribution" title="Hyperexponential distribution">Hyperexponential</a></li> <li><a href="/wiki/Hypoexponential_distribution" title="Hypoexponential distribution">Hypoexponential</a></li> <li><a href="/wiki/Exponential-logarithmic_distribution" title="Exponential-logarithmic distribution">Logarithmic</a></li></ul></li> <li><a href="/wiki/F-distribution" title="F-distribution"><i>F</i></a> <ul><li><a href="/wiki/Noncentral_F-distribution" title="Noncentral F-distribution">Noncentral</a></li></ul></li> <li><a href="/wiki/Folded_normal_distribution" title="Folded normal distribution">Folded normal</a></li> <li><a href="/wiki/Fr%C3%A9chet_distribution" title="Fréchet distribution">Fréchet</a></li> <li><a href="/wiki/Gamma_distribution" title="Gamma distribution">Gamma</a> <ul><li><a href="/wiki/Generalized_gamma_distribution" title="Generalized gamma distribution">Generalized</a></li> <li><a href="/wiki/Inverse-gamma_distribution" title="Inverse-gamma distribution">Inverse</a></li></ul></li> <li><a href="/wiki/Gamma/Gompertz_distribution" title="Gamma/Gompertz distribution">gamma/Gompertz</a></li> <li><a href="/wiki/Gompertz_distribution" title="Gompertz distribution">Gompertz</a> <ul><li><a href="/wiki/Shifted_Gompertz_distribution" title="Shifted Gompertz distribution">Shifted</a></li></ul></li> <li><a href="/wiki/Half-logistic_distribution" title="Half-logistic distribution">Half-logistic</a></li> <li><a href="/wiki/Half-normal_distribution" title="Half-normal distribution">Half-normal</a></li> <li><a href="/wiki/Hotelling%27s_T-squared_distribution" title="Hotelling's T-squared distribution">Hotelling's <i>T</i>-squared</a></li> <li><a href="/wiki/Inverse_Gaussian_distribution" title="Inverse Gaussian distribution">Inverse Gaussian</a> <ul><li><a href="/wiki/Generalized_inverse_Gaussian_distribution" title="Generalized inverse Gaussian distribution">Generalized</a></li></ul></li> <li><a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov</a></li> <li><a href="/wiki/L%C3%A9vy_distribution" title="Lévy distribution">Lévy</a></li> <li><a href="/wiki/Log-Cauchy_distribution" title="Log-Cauchy distribution">Log-Cauchy</a></li> <li><a href="/wiki/Log-Laplace_distribution" title="Log-Laplace distribution">Log-Laplace</a></li> <li><a href="/wiki/Log-logistic_distribution" title="Log-logistic distribution">Log-logistic</a></li> <li><a href="/wiki/Log-normal_distribution" title="Log-normal distribution">Log-normal</a></li> <li><a href="/wiki/Log-t_distribution" title="Log-t distribution">Log-t</a></li> <li><a href="/wiki/Lomax_distribution" title="Lomax distribution">Lomax</a></li> <li><a href="/wiki/Matrix-exponential_distribution" title="Matrix-exponential distribution">Matrix-exponential</a></li> <li><a href="/wiki/Maxwell%E2%80%93Boltzmann_distribution" title="Maxwell–Boltzmann distribution">Maxwell–Boltzmann</a></li> <li><a href="/wiki/Maxwell%E2%80%93J%C3%BCttner_distribution" title="Maxwell–Jüttner distribution">Maxwell–Jüttner</a></li> <li><a href="/wiki/Mittag-Leffler_distribution" title="Mittag-Leffler distribution">Mittag-Leffler</a></li> <li><a href="/wiki/Nakagami_distribution" title="Nakagami distribution">Nakagami</a></li> <li><a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto</a></li> <li><a href="/wiki/Phase-type_distribution" title="Phase-type distribution">Phase-type</a></li> <li><a href="/wiki/Poly-Weibull_distribution" title="Poly-Weibull distribution">Poly-Weibull</a></li> <li><a href="/wiki/Rayleigh_distribution" title="Rayleigh distribution">Rayleigh</a></li> <li><a href="/wiki/Relativistic_Breit%E2%80%93Wigner_distribution" title="Relativistic Breit–Wigner distribution">Relativistic Breit–Wigner</a></li> <li><a href="/wiki/Rice_distribution" title="Rice distribution">Rice</a></li> <li><a href="/wiki/Truncated_normal_distribution" title="Truncated normal distribution">Truncated normal</a></li> <li><a href="/wiki/Type-2_Gumbel_distribution" title="Type-2 Gumbel distribution">type-2 Gumbel</a></li> <li><a href="/wiki/Weibull_distribution" title="Weibull distribution">Weibull</a> <ul><li><a href="/wiki/Discrete_Weibull_distribution" title="Discrete Weibull distribution">Discrete</a></li></ul></li> <li><a href="/wiki/Wilks%27s_lambda_distribution" title="Wilks's lambda distribution">Wilks's lambda</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">supported <br />on the whole <br />real line</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cauchy_distribution" title="Cauchy distribution">Cauchy</a></li> <li><a href="/wiki/Generalized_normal_distribution#Version_1" title="Generalized normal distribution">Exponential power</a></li> <li><a href="/wiki/Fisher%27s_z-distribution" title="Fisher's z-distribution">Fisher's <i>z</i></a></li> <li><a href="/wiki/Kaniadakis_Gaussian_distribution" title="Kaniadakis Gaussian distribution">Kaniadakis κ-Gaussian</a></li> <li><a href="/wiki/Gaussian_q-distribution" title="Gaussian q-distribution">Gaussian <i>q</i></a></li> <li><a href="/wiki/Generalized_normal_distribution" title="Generalized normal distribution">Generalized normal</a></li> <li><a href="/wiki/Generalised_hyperbolic_distribution" title="Generalised hyperbolic distribution">Generalized hyperbolic</a></li> <li><a href="/wiki/Geometric_stable_distribution" title="Geometric stable distribution">Geometric stable</a></li> <li><a href="/wiki/Gumbel_distribution" title="Gumbel distribution">Gumbel</a></li> <li><a href="/wiki/Holtsmark_distribution" title="Holtsmark distribution">Holtsmark</a></li> <li><a href="/wiki/Hyperbolic_secant_distribution" title="Hyperbolic secant distribution">Hyperbolic secant</a></li> <li><a href="/wiki/Johnson%27s_SU-distribution" title="Johnson's SU-distribution">Johnson's <i>S<sub>U</sub></i></a></li> <li><a href="/wiki/Landau_distribution" title="Landau distribution">Landau</a></li> <li><a href="/wiki/Laplace_distribution" title="Laplace distribution">Laplace</a> <ul><li><a href="/wiki/Asymmetric_Laplace_distribution" title="Asymmetric Laplace distribution">Asymmetric</a></li></ul></li> <li><a href="/wiki/Logistic_distribution" title="Logistic distribution">Logistic</a></li> <li><a href="/wiki/Noncentral_t-distribution" title="Noncentral t-distribution">Noncentral <i>t</i></a></li> <li><a href="/wiki/Normal_distribution" title="Normal distribution">Normal (Gaussian)</a></li> <li><a href="/wiki/Normal-inverse_Gaussian_distribution" title="Normal-inverse Gaussian distribution">Normal-inverse Gaussian</a></li> <li><a href="/wiki/Skew_normal_distribution" title="Skew normal distribution">Skew normal</a></li> <li><a href="/wiki/Slash_distribution" title="Slash distribution">Slash</a></li> <li><a href="/wiki/Stable_distribution" title="Stable distribution">Stable</a></li> <li><a href="/wiki/Student%27s_t-distribution" title="Student's t-distribution">Student's <i>t</i></a></li> <li><a href="/wiki/Tracy%E2%80%93Widom_distribution" title="Tracy–Widom distribution">Tracy–Widom</a></li> <li><a href="/wiki/Variance-gamma_distribution" title="Variance-gamma distribution">Variance-gamma</a></li> <li><a href="/wiki/Voigt_profile" title="Voigt profile">Voigt</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">with support <br />whose type varies</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Generalized_chi-squared_distribution" title="Generalized chi-squared distribution">Generalized chi-squared</a></li> <li><a href="/wiki/Generalized_extreme_value_distribution" title="Generalized extreme value distribution">Generalized extreme value</a></li> <li><a href="/wiki/Generalized_Pareto_distribution" title="Generalized Pareto distribution">Generalized Pareto</a></li> <li><a href="/wiki/Marchenko%E2%80%93Pastur_distribution" title="Marchenko–Pastur distribution">Marchenko–Pastur</a></li> <li><a href="/wiki/Kaniadakis_Exponential_distribution" class="mw-redirect" title="Kaniadakis Exponential distribution">Kaniadakis <i>κ</i>-exponential</a></li> <li><a href="/wiki/Kaniadakis_Gamma_distribution" title="Kaniadakis Gamma distribution">Kaniadakis <i>κ</i>-Gamma</a></li> <li><a href="/wiki/Kaniadakis_Weibull_distribution" title="Kaniadakis Weibull distribution">Kaniadakis <i>κ</i>-Weibull</a></li> <li><a href="/wiki/Kaniadakis_Logistic_distribution" class="mw-redirect" title="Kaniadakis Logistic distribution">Kaniadakis <i>κ</i>-Logistic</a></li> <li><a href="/wiki/Kaniadakis_Erlang_distribution" title="Kaniadakis Erlang distribution">Kaniadakis <i>κ</i>-Erlang</a></li> <li><a href="/wiki/Q-exponential_distribution" title="Q-exponential distribution"><i>q</i>-exponential</a></li> <li><a href="/wiki/Q-Gaussian_distribution" title="Q-Gaussian distribution"><i>q</i>-Gaussian</a></li> <li><a href="/wiki/Q-Weibull_distribution" title="Q-Weibull distribution"><i>q</i>-Weibull</a></li> <li><a href="/wiki/Shifted_log-logistic_distribution" title="Shifted log-logistic distribution">Shifted log-logistic</a></li> <li><a href="/wiki/Tukey_lambda_distribution" title="Tukey lambda distribution">Tukey lambda</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mixed <br />univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">continuous-<br />discrete</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Rectified_Gaussian_distribution" title="Rectified Gaussian distribution">Rectified Gaussian</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Joint_probability_distribution" title="Joint probability distribution">Multivariate <br />(joint)</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><span class="nobold"><i>Discrete: </i></span></li> <li><a href="/wiki/Ewens%27s_sampling_formula" title="Ewens's sampling formula">Ewens</a></li> <li><a href="/wiki/Multinomial_distribution" title="Multinomial distribution">Multinomial</a> <ul><li><a href="/wiki/Dirichlet-multinomial_distribution" title="Dirichlet-multinomial distribution">Dirichlet</a></li> <li><a href="/wiki/Negative_multinomial_distribution" title="Negative multinomial distribution">Negative</a></li></ul></li> <li><span class="nobold"><i>Continuous: </i></span></li> <li><a href="/wiki/Dirichlet_distribution" title="Dirichlet distribution">Dirichlet</a> <ul><li><a href="/wiki/Generalized_Dirichlet_distribution" title="Generalized Dirichlet distribution">Generalized</a></li></ul></li> <li><a href="/wiki/Multivariate_Laplace_distribution" title="Multivariate Laplace distribution">Multivariate Laplace</a></li> <li><a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">Multivariate normal</a></li> <li><a href="/wiki/Multivariate_stable_distribution" title="Multivariate stable distribution">Multivariate stable</a></li> <li><a href="/wiki/Multivariate_t-distribution" title="Multivariate t-distribution">Multivariate <i>t</i></a></li> <li><a href="/wiki/Normal-gamma_distribution" title="Normal-gamma distribution">Normal-gamma</a> <ul><li><a href="/wiki/Normal-inverse-gamma_distribution" title="Normal-inverse-gamma distribution">Inverse</a></li></ul></li> <li><span class="nobold"><i><a href="/wiki/Random_matrix" title="Random matrix">Matrix-valued: </a></i></span></li> <li><a href="/wiki/Lewandowski-Kurowicka-Joe_distribution" title="Lewandowski-Kurowicka-Joe distribution">LKJ</a></li> <li><a href="/wiki/Matrix_normal_distribution" title="Matrix normal distribution">Matrix normal</a></li> <li><a href="/wiki/Matrix_t-distribution" title="Matrix t-distribution">Matrix <i>t</i></a></li> <li><a href="/wiki/Matrix_gamma_distribution" title="Matrix gamma distribution">Matrix gamma</a> <ul><li><a href="/wiki/Inverse_matrix_gamma_distribution" title="Inverse matrix gamma distribution">Inverse</a></li></ul></li> <li><a href="/wiki/Wishart_distribution" title="Wishart distribution">Wishart</a> <ul><li><a href="/wiki/Normal-Wishart_distribution" title="Normal-Wishart distribution">Normal</a></li> <li><a href="/wiki/Inverse-Wishart_distribution" title="Inverse-Wishart distribution">Inverse</a></li> <li><a href="/wiki/Normal-inverse-Wishart_distribution" title="Normal-inverse-Wishart distribution">Normal-inverse</a></li> <li><a href="/wiki/Complex_Wishart_distribution" title="Complex Wishart distribution">Complex</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Directional_statistics" title="Directional statistics">Directional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <dl><dt><span class="nobold"><i>Univariate (circular) <a href="/wiki/Directional_statistics" title="Directional statistics">directional</a></i></span></dt> <dd><a href="/wiki/Circular_uniform_distribution" title="Circular uniform distribution">Circular uniform</a></dd> <dd><a href="/wiki/Von_Mises_distribution" title="Von Mises distribution">Univariate von Mises</a></dd> <dd><a href="/wiki/Wrapped_normal_distribution" title="Wrapped normal distribution">Wrapped normal</a></dd> <dd><a href="/wiki/Wrapped_Cauchy_distribution" title="Wrapped Cauchy distribution">Wrapped Cauchy</a></dd> <dd><a href="/wiki/Wrapped_exponential_distribution" title="Wrapped exponential distribution">Wrapped exponential</a></dd> <dd><a href="/wiki/Wrapped_asymmetric_Laplace_distribution" title="Wrapped asymmetric Laplace distribution">Wrapped asymmetric Laplace</a></dd> <dd><a href="/wiki/Wrapped_L%C3%A9vy_distribution" title="Wrapped Lévy distribution">Wrapped Lévy</a></dd> <dt><span class="nobold"><i>Bivariate (spherical)</i></span></dt> <dd><a href="/wiki/Kent_distribution" title="Kent distribution">Kent</a></dd> <dt><span class="nobold"><i>Bivariate (toroidal)</i></span></dt> <dd><a href="/wiki/Bivariate_von_Mises_distribution" title="Bivariate von Mises distribution">Bivariate von Mises</a></dd> <dt><span class="nobold"><i>Multivariate</i></span></dt> <dd><a href="/wiki/Von_Mises%E2%80%93Fisher_distribution" title="Von Mises–Fisher distribution">von Mises–Fisher</a></dd> <dd><a href="/wiki/Bingham_distribution" title="Bingham distribution">Bingham</a></dd></dl> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Degenerate_distribution" title="Degenerate distribution">Degenerate</a> <br />and <a href="/wiki/Singular_distribution" title="Singular distribution">singular</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <dl><dt><span class="nobold"><i>Degenerate</i></span></dt> <dd><a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a></dd> <dt><span class="nobold"><i>Singular</i></span></dt> <dd><a href="/wiki/Cantor_distribution" title="Cantor distribution">Cantor</a></dd></dl> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Families</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Circular_distribution" title="Circular distribution">Circular</a></li> <li><a href="/wiki/Compound_Poisson_distribution" title="Compound Poisson distribution">Compound Poisson</a></li> <li><a href="/wiki/Elliptical_distribution" title="Elliptical distribution">Elliptical</a></li> <li><a href="/wiki/Exponential_family" title="Exponential family">Exponential</a></li> <li><a href="/wiki/Natural_exponential_family" title="Natural exponential family">Natural exponential</a></li> <li><a href="/wiki/Location%E2%80%93scale_family" title="Location–scale family">Location–scale</a></li> <li><a href="/wiki/Maximum_entropy_probability_distribution" title="Maximum entropy probability distribution">Maximum entropy</a></li> <li><a href="/wiki/Mixture_distribution" title="Mixture distribution">Mixture</a></li> <li><a href="/wiki/Pearson_distribution" title="Pearson distribution">Pearson</a></li> <li><a href="/wiki/Tweedie_distribution" title="Tweedie distribution">Tweedie</a></li> <li><a href="/wiki/Wrapped_distribution" title="Wrapped distribution">Wrapped</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span 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Hypergeometric distribution - Wikipedia