Birthday problem - Wikipedia

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vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Calculating_the_probability" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Calculating_the_probability"> <div class="vector-toc-text"> 1 Calculating the probability </div> </a> <ul id="toc-Calculating_the_probability-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Approximations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Approximations"> <div class="vector-toc-text"> 2 Approximations </div> </a> <button aria-controls="toc-Approximations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Approximations subsection </button> <ul id="toc-Approximations-sublist" class="vector-toc-list"> <li id="toc-Simple_exponentiation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simple_exponentiation"> <div class="vector-toc-text"> 2.1 Simple exponentiation </div> </a> <ul id="toc-Simple_exponentiation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Poisson_approximation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Poisson_approximation"> <div class="vector-toc-text"> 2.2 Poisson approximation </div> </a> <ul id="toc-Poisson_approximation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Square_approximation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Square_approximation"> <div class="vector-toc-text"> 2.3 Square approximation </div> </a> <ul id="toc-Square_approximation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Approximation_of_number_of_people" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Approximation_of_number_of_people"> <div class="vector-toc-text"> 2.4 Approximation of number of people </div> </a> <ul id="toc-Approximation_of_number_of_people-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Probability_table" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Probability_table"> <div class="vector-toc-text"> 2.5 Probability table </div> </a> <ul id="toc-Probability_table-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-An_upper_bound_on_the_probability_and_a_lower_bound_on_the_number_of_people" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#An_upper_bound_on_the_probability_and_a_lower_bound_on_the_number_of_people"> <div class="vector-toc-text"> 3 An upper bound on the probability and a lower bound on the number of people </div> </a> <ul id="toc-An_upper_bound_on_the_probability_and_a_lower_bound_on_the_number_of_people-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> 4 Generalizations </div> </a> <button aria-controls="toc-Generalizations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Generalizations subsection </button> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> <li id="toc-Arbitrary_number_of_days" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Arbitrary_number_of_days"> <div class="vector-toc-text"> 4.1 Arbitrary number of days </div> </a> <ul id="toc-Arbitrary_number_of_days-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-More_than_two_people_sharing_a_birthday" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#More_than_two_people_sharing_a_birthday"> <div class="vector-toc-text"> 4.2 More than two people sharing a birthday </div> </a> <ul id="toc-More_than_two_people_sharing_a_birthday-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Everyone_shares_a_birthday" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Everyone_shares_a_birthday"> <div class="vector-toc-text"> 4.3 Everyone shares a birthday </div> </a> <ul id="toc-Everyone_shares_a_birthday-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Probability_of_a_shared_birthday_(collision)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Probability_of_a_shared_birthday_(collision)"> <div class="vector-toc-text"> 4.4 Probability of a shared birthday (collision) </div> </a> <ul id="toc-Probability_of_a_shared_birthday_(collision)-sublist" class="vector-toc-list"> <li id="toc-Probability_of_a_unique_collision" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Probability_of_a_unique_collision"> <div class="vector-toc-text"> 4.4.1 Probability of a unique collision </div> </a> <ul id="toc-Probability_of_a_unique_collision-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalization_to_multiple_types_of_people" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Generalization_to_multiple_types_of_people"> <div class="vector-toc-text"> 4.4.2 Generalization to multiple types of people </div> </a> <ul id="toc-Generalization_to_multiple_types_of_people-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Other_birthday_problems" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Other_birthday_problems"> <div class="vector-toc-text"> 5 Other birthday problems </div> </a> <button aria-controls="toc-Other_birthday_problems-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Other birthday problems subsection </button> <ul id="toc-Other_birthday_problems-sublist" class="vector-toc-list"> <li id="toc-First_match" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#First_match"> <div class="vector-toc-text"> 5.1 First match </div> </a> <ul id="toc-First_match-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Same_birthday_as_you" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Same_birthday_as_you"> <div class="vector-toc-text"> 5.2 Same birthday as you </div> </a> <ul id="toc-Same_birthday_as_you-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Number_of_people_with_a_shared_birthday" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Number_of_people_with_a_shared_birthday"> <div class="vector-toc-text"> 5.3 Number of people with a shared birthday </div> </a> <ul id="toc-Number_of_people_with_a_shared_birthday-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Number_of_people_until_every_birthday_is_achieved" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Number_of_people_until_every_birthday_is_achieved"> <div class="vector-toc-text"> 5.4 Number of people until every birthday is achieved </div> </a> <ul id="toc-Number_of_people_until_every_birthday_is_achieved-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Near_matches" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Near_matches"> <div class="vector-toc-text"> 5.5 Near matches </div> </a> <ul id="toc-Near_matches-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Number_of_days_with_a_certain_number_of_birthdays" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Number_of_days_with_a_certain_number_of_birthdays"> <div class="vector-toc-text"> 5.6 Number of days with a certain number of birthdays </div> </a> <ul id="toc-Number_of_days_with_a_certain_number_of_birthdays-sublist" class="vector-toc-list"> <li id="toc-Number_of_days_with_at_least_one_birthday" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Number_of_days_with_at_least_one_birthday"> <div class="vector-toc-text"> 5.6.1 Number of days with at least one birthday </div> </a> <ul id="toc-Number_of_days_with_at_least_one_birthday-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Number_of_days_with_at_least_two_birthdays" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Number_of_days_with_at_least_two_birthdays"> <div class="vector-toc-text"> 5.6.2 Number of days with at least two birthdays </div> </a> <ul id="toc-Number_of_days_with_at_least_two_birthdays-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Number_of_people_who_repeat_a_birthday" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Number_of_people_who_repeat_a_birthday"> <div class="vector-toc-text"> 5.7 Number of people who repeat a birthday </div> </a> <ul id="toc-Number_of_people_who_repeat_a_birthday-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Average_number_of_people_to_get_at_least_one_shared_birthday" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Average_number_of_people_to_get_at_least_one_shared_birthday"> <div class="vector-toc-text"> 5.8 Average number of people to get at least one shared birthday </div> </a> <ul id="toc-Average_number_of_people_to_get_at_least_one_shared_birthday-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reverse_problem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Reverse_problem"> <div class="vector-toc-text"> 5.9 Reverse problem </div> </a> <ul id="toc-Reverse_problem-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Partition_problem" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Partition_problem"> <div class="vector-toc-text"> 6 Partition problem </div> </a> <ul id="toc-Partition_problem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_fiction" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#In_fiction"> <div class="vector-toc-text"> 7 In fiction </div> </a> <ul id="toc-In_fiction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 8 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 9 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> 10 Bibliography </div> </a> <ul id="toc-Bibliography-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 11 External links </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" title="Table of Contents" > <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" > Toggle the table of contents </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">Birthday problem</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 39 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-39" aria-hidden="true" > 39 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B9%D8%B6%D9%84%D8%A9_%D9%8A%D9%88%D9%85_%D8%A7%D9%84%D9%85%D9%8A%D9%84%D8%A7%D8%AF" title="معضلة يوم الميلاد – Arabic" lang="ar" hreflang="ar" data-title="معضلة يوم الميلاد" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Ro%C4%91endanski_paradoks" title="Rođendanski paradoks – Bosnian" lang="bs" hreflang="bs" data-title="Rođendanski paradoks" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Problema_dels_aniversaris" title="Problema dels aniversaris – Catalan" lang="ca" hreflang="ca" data-title="Problema dels aniversaris" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Narozeninov%C3%BD_probl%C3%A9m" title="Narozeninový problém – Czech" lang="cs" hreflang="cs" data-title="Narozeninový problém" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/F%C3%B8dselsdagsparadokset" title="Fødselsdagsparadokset – Danish" lang="da" hreflang="da" data-title="Fødselsdagsparadokset" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Geburtstagsparadoxon" title="Geburtstagsparadoxon – German" lang="de" hreflang="de" data-title="Geburtstagsparadoxon" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CE%B1%CF%81%CE%AC%CE%B4%CE%BF%CE%BE%CE%BF_%CF%84%CF%89%CE%BD_%CE%B3%CE%B5%CE%BD%CE%B5%CE%B8%CE%BB%CE%AF%CF%89%CE%BD" title="Παράδοξο των γενεθλίων – Greek" lang="el" hreflang="el" data-title="Παράδοξο των γενεθλίων" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Paradoja_del_cumplea%C3%B1os" title="Paradoja del cumpleaños – Spanish" lang="es" hreflang="es" data-title="Paradoja del cumpleaños" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Urtebetetzeen_ebazkizuna" title="Urtebetetzeen ebazkizuna – Basque" lang="eu" hreflang="eu" data-title="Urtebetetzeen ebazkizuna" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%B3%D8%A6%D9%84%D9%87_%D8%AA%D8%A7%D8%B1%DB%8C%D8%AE_%D8%AA%D9%88%D9%84%D8%AF" title="مسئله تاریخ تولد – Persian" lang="fa" hreflang="fa" data-title="مسئله تاریخ تولد" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Paradoxe_des_anniversaires" title="Paradoxe des anniversaires – French" lang="fr" hreflang="fr" data-title="Paradoxe des anniversaires" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Paradoxo_do_aniversario" title="Paradoxo do aniversario – Galician" lang="gl" hreflang="gl" data-title="Paradoxo do aniversario" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%83%9D%EC%9D%BC_%EB%AC%B8%EC%A0%9C" title="생일 문제 – Korean" lang="ko" hreflang="ko" data-title="생일 문제" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BE%D5%B6%D5%B6%D5%A4%D5%B5%D5%A1%D5%B6_%D6%85%D6%80%D5%A5%D6%80%D5%AB_%D5%AD%D5%B6%D5%A4%D5%AB%D6%80" title="Ծննդյան օրերի խնդիր – Armenian" lang="hy" hreflang="hy" data-title="Ծննդյան օրերի խնդիր" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Ro%C4%91endanski_problem" title="Rođendanski problem – Croatian" lang="hr" hreflang="hr" data-title="Rođendanski problem" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Paradosso_del_compleanno" title="Paradosso del compleanno – Italian" lang="it" hreflang="it" data-title="Paradosso del compleanno" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%A8%D7%93%D7%95%D7%A7%D7%A1_%D7%99%D7%95%D7%9D_%D7%94%D7%94%D7%95%D7%9C%D7%93%D7%AA" title="פרדוקס יום ההולדת – Hebrew" lang="he" hreflang="he" data-title="פרדוקס יום ההולדת" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Problema_natalium" title="Problema natalium – Latin" lang="la" hreflang="la" data-title="Problema natalium" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Gimimo_dien%C5%B3_paradoksas" title="Gimimo dienų paradoksas – Lithuanian" lang="lt" hreflang="lt" data-title="Gimimo dienų paradoksas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Sz%C3%BClet%C3%A9snap-paradoxon" title="Születésnap-paradoxon – Hungarian" lang="hu" hreflang="hu" data-title="Születésnap-paradoxon" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%9C%E0%B4%A8%E0%B5%8D%E0%B4%AE%E0%B4%A6%E0%B4%BF%E0%B4%A8%E0%B4%AA%E0%B5%8D%E0%B4%B0%E0%B4%B6%E0%B5%8D%E0%B4%A8%E0%B4%82" title="ജന്മദിനപ്രശ്നം – Malayalam" lang="ml" hreflang="ml" data-title="ജന്മദിനപ്രശ്നം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BE%E0%A4%A2%E0%A4%A6%E0%A4%BF%E0%A4%B5%E0%A4%B8_%E0%A4%B8%E0%A4%AE%E0%A4%B8%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="वाढदिवस समस्या – Marathi" lang="mr" hreflang="mr" data-title="वाढदिवस समस्या" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Verjaardagenparadox" title="Verjaardagenparadox – Dutch" lang="nl" hreflang="nl" data-title="Verjaardagenparadox" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%AA%95%E7%94%9F%E6%97%A5%E3%81%AE%E3%83%91%E3%83%A9%E3%83%89%E3%83%83%E3%82%AF%E3%82%B9" title="誕生日のパラドックス – Japanese" lang="ja" hreflang="ja" data-title="誕生日のパラドックス" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Parad%C3%B2xa_dels_anniversaris" title="Paradòxa dels anniversaris – Occitan" lang="oc" hreflang="oc" data-title="Paradòxa dels anniversaris" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Paradoks_dni_urodzin" title="Paradoks dni urodzin – Polish" lang="pl" hreflang="pl" data-title="Paradoks dni urodzin" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Paradoxo_do_anivers%C3%A1rio" title="Paradoxo do aniversário – Portuguese" lang="pt" hreflang="pt" data-title="Paradoxo do aniversário" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%B4%D0%BE%D0%BA%D1%81_%D0%B4%D0%BD%D0%B5%D0%B9_%D1%80%D0%BE%D0%B6%D0%B4%D0%B5%D0%BD%D0%B8%D1%8F" title="Парадокс дней рождения – Russian" lang="ru" hreflang="ru" data-title="Парадокс дней рождения" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Problemi_i_dit%C3%ABlindjes" title="Problemi i ditëlindjes – 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class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle">(Redirected from <a href="/w/index.php?title=Birthday_paradox&redirect=no" class="mw-redirect" title="Birthday paradox">Birthday paradox</a>)</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">Probability of shared birthdays</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">For yearly variation in mortality rates, see <a href="/wiki/Birthday_effect" title="Birthday effect">Birthday effect</a>. For the mathematical brain teaser that was asked in the Math Olympiad, see <a href="/wiki/Cheryl%27s_Birthday" title="Cheryl's Birthday">Cheryl's Birthday</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Birthday_Paradox.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Birthday_Paradox.svg/290px-Birthday_Paradox.svg.png" decoding="async" width="290" height="187" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Birthday_Paradox.svg/435px-Birthday_Paradox.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Birthday_Paradox.svg/580px-Birthday_Paradox.svg.png 2x" data-file-width="1289" data-file-height="830" /></a><figcaption>The computed probability of at least two people sharing the same birthday versus the number of people</figcaption></figure> In <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, the birthday problem asks for the probability that, in a set of n <a href="/wiki/Random" class="mw-redirect" title="Random">randomly</a> chosen people, at least two will share the same <a href="/wiki/Birthday" title="Birthday">birthday</a>. The birthday paradox is the counterintuitive fact that only 23 people are needed for that probability to exceed 50%. The birthday paradox is a <a href="/wiki/Veridical_paradox" class="mw-redirect" title="Veridical paradox">veridical paradox</a>: it seems wrong at first glance but is, in fact, true. While it may seem surprising that only 23 individuals are required to reach a 50% probability of a shared birthday, this result is made more intuitive by considering that the birthday comparisons will be made between every possible pair of individuals. With 23 individuals, there are <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠23 × 22/2⁠ = 253 pairs to consider. Real-world applications for the birthday problem include a cryptographic attack called the <a href="/wiki/Birthday_attack" title="Birthday attack">birthday attack</a>, which uses this probabilistic model to reduce the complexity of finding a <a href="/wiki/Collision_attack" title="Collision attack">collision</a> for a <a href="/wiki/Hash_function" title="Hash function">hash function</a>, as well as calculating the approximate risk of a hash collision existing within the hashes of a given size of population. The problem is generally attributed to <a href="/wiki/Harold_Davenport" title="Harold Davenport">Harold Davenport</a> in about 1927, though he did not publish it at the time. Davenport did not claim to be its discoverer "because he could not believe that it had not been stated earlier".<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a> The first publication of a version of the birthday problem was by <a href="/wiki/Richard_von_Mises" title="Richard von Mises">Richard von Mises</a> in 1939.<a href="#cite_note-3">[3]</a> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Calculating_the_probability">Calculating the probability</h2>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=1" title="Edit section: Calculating the probability">edit</a>]</div> From a <a href="/wiki/Permutation" title="Permutation">permutations</a> perspective, let the event A be the probability of finding a group of 23 people without any repeated birthdays. Where the event B is the probability of finding a group of 23 people with at least two people sharing same birthday, P(B) = 1 − P(A). This is such that P(A) is the ratio of the total number of birthdays, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{nr}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{nr}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66d968dc25fe4ec5317655346ff7214b81b21fd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.315ex; height:2.509ex;" alt="{\displaystyle V_{nr}}" />, without repetitions and order matters (e.g. for a group of 2 people, mm/dd birthday format, one possible outcome is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{\left\{01/02,05/20\right\},\left\{05/20,01/02\right\},\left\{10/02,08/04\right\},...\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mrow> <mo>{</mo> <mrow> <mn>01</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>02</mn> <mo>,</mo> <mn>05</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>20</mn> </mrow> <mo>}</mo> </mrow> <mo>,</mo> <mrow> <mo>{</mo> <mrow> <mn>05</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>20</mn> <mo>,</mo> <mn>01</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>02</mn> </mrow> <mo>}</mo> </mrow> <mo>,</mo> <mrow> <mo>{</mo> <mrow> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>02</mn> <mo>,</mo> <mn>08</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>04</mn> </mrow> <mo>}</mo> </mrow> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{\left\{01/02,05/20\right\},\left\{05/20,01/02\right\},\left\{10/02,08/04\right\},...\right\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1af65822c4e7ee0d37f977243e4f1d7730ec8970" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.64ex; height:2.843ex;" alt="{\displaystyle \left\{\left\{01/02,05/20\right\},\left\{05/20,01/02\right\},\left\{10/02,08/04\right\},...\right\}}" />) divided by the total number of birthdays with repetition and order matters, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27b61a6ac590c5cd1911b10c484f38de1edb58c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.181ex; height:2.509ex;" alt="{\displaystyle V_{t}}" />, as it is the total space of outcomes from the experiment (e.g. 2 people, one possible outcome is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{\left\{01/02,01/02\right\},\left\{10/02,08/04\right\},...\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mrow> <mo>{</mo> <mrow> <mn>01</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>02</mn> <mo>,</mo> <mn>01</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>02</mn> </mrow> <mo>}</mo> </mrow> <mo>,</mo> <mrow> <mo>{</mo> <mrow> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>02</mn> <mo>,</mo> <mn>08</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>04</mn> </mrow> <mo>}</mo> </mrow> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{\left\{01/02,01/02\right\},\left\{10/02,08/04\right\},...\right\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b41ca151f4e0564f7a6613292dca65444d73c18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.236ex; height:2.843ex;" alt="{\displaystyle \left\{\left\{01/02,01/02\right\},\left\{10/02,08/04\right\},...\right\}}" />). Therefore <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{nr}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{nr}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66d968dc25fe4ec5317655346ff7214b81b21fd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.315ex; height:2.509ex;" alt="{\displaystyle V_{nr}}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27b61a6ac590c5cd1911b10c484f38de1edb58c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.181ex; height:2.509ex;" alt="{\displaystyle V_{t}}" /> are <a href="/wiki/Permutation" title="Permutation">permutations</a>. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}V_{nr}&={\frac {n!}{(n-k)!}}={\frac {365!}{(365-23)!}}\\[8pt]V_{t}&=n^{k}=365^{23}\\[8pt]P(A)&={\frac {V_{nr}}{V_{t}}}\approx 0.492703\\[8pt]P(B)&=1-P(A)\approx 1-0.492703\approx 0.507297\quad (50.7297\%)\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 1.1em 1.1em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>r</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>365</mn> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>365</mn> <mo>−</mo> <mn>23</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <msup> <mn>365</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>r</mi> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mfrac> </mrow> <mo>≈</mo> <mn>0.492703</mn> </mtd> </mtr> <mtr> <mtd> <mi>P</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mn>1</mn> <mo>−</mo> <mn>0.492703</mn> <mo>≈</mo> <mn>0.507297</mn> <mspace width="1em"></mspace> <mo stretchy="false">(</mo> <mn>50.7297</mn> <mi mathvariant="normal">%</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}V_{nr}&={\frac {n!}{(n-k)!}}={\frac {365!}{(365-23)!}}\\[8pt]V_{t}&=n^{k}=365^{23}\\[8pt]P(A)&={\frac {V_{nr}}{V_{t}}}\approx 0.492703\\[8pt]P(B)&=1-P(A)\approx 1-0.492703\approx 0.507297\quad (50.7297\%)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8b14f49cb83d4079d928a6aff2c3164c2e8539" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.338ex; width:59.927ex; height:23.843ex;" alt="{\displaystyle {\begin{aligned}V_{nr}&={\frac {n!}{(n-k)!}}={\frac {365!}{(365-23)!}}\\[8pt]V_{t}&=n^{k}=365^{23}\\[8pt]P(A)&={\frac {V_{nr}}{V_{t}}}\approx 0.492703\\[8pt]P(B)&=1-P(A)\approx 1-0.492703\approx 0.507297\quad (50.7297\%)\end{aligned}}}" /></dd></dl> Another way the birthday problem can be solved is by asking for an approximate probability that in a group of n people at least two have the same birthday. For simplicity, <a href="/wiki/Leap_year" title="Leap year">leap years</a>, <a href="/wiki/Twin" title="Twin">twins</a>, <a href="/wiki/Selection_bias" title="Selection bias">selection bias</a>, and seasonal and weekly variations in birth rates<a href="#cite_note-4">[4]</a> are generally disregarded, and instead it is assumed that there are 365 possible birthdays, and that each person's birthday is equally likely to be any of these days, independent of the other people in the group. For independent birthdays, a <a href="/wiki/Discrete_uniform_distribution" title="Discrete uniform distribution">uniform distribution</a> of birthdays minimizes the probability of two people in a group having the same birthday. Any unevenness increases the likelihood of two people sharing a birthday.<a href="#cite_note-5">[5]</a><a href="#cite_note-6">[6]</a> However real-world birthdays are not sufficiently uneven to make much change: the real-world group size necessary to have a greater than 50% chance of a shared birthday is 23, as in the theoretical uniform distribution.<a href="#cite_note-Borja-7">[7]</a> The goal is to compute P(B), the probability that at least two people in the room have the same birthday. However, it is simpler to calculate P(A′), the probability that no two people in the room have the same birthday. Then, because B and A′ are the only two possibilities and are also <a href="/wiki/Mutually_exclusive_events" class="mw-redirect" title="Mutually exclusive events">mutually exclusive</a>, P(B) = 1 − P(A′). Here is the calculation of P(B) for 23 people. Let the 23 people be numbered 1 to 23. The <a href="/wiki/Event_(probability_theory)" title="Event (probability theory)">event</a> that all 23 people have different birthdays is the same as the event that person 2 does not have the same birthday as person 1, and that person 3 does not have the same birthday as either person 1 or person 2, and so on, and finally that person 23 does not have the same birthday as any of persons 1 through 22. Let these events be called Event 2, Event 3, and so on. Event 1 is the event of person 1 having a birthday, which occurs with probability 1. This conjunction of events may be computed using <a href="/wiki/Conditional_probability" title="Conditional probability">conditional probability</a>: the probability of Event 2 is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠364/365⁠, as person 2 may have any birthday other than the birthday of person 1. Similarly, the probability of Event 3 given that Event 2 occurred is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠363/365⁠, as person 3 may have any of the birthdays not already taken by persons 1 and 2. This continues until finally the probability of Event 23 given that all preceding events occurred is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠343/365⁠. Finally, the principle of conditional probability implies that P(A′) is equal to the product of these individual probabilities: <style data-mw-deduplicate="TemplateStyles:r1266403038">.mw-parser-output table.numblk{border-collapse:collapse;border:none;margin-top:0;margin-right:0;margin-bottom:0}.mw-parser-output table.numblk>tbody>tr>td{vertical-align:middle;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2){width:99%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table{border-collapse:collapse;margin:0;border:none;width:100%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:first-child,.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:last-child{padding:0 0.4ex}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:nth-child(2){width:100%;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{padding:0}.mw-parser-output table.numblk>tbody>tr>td:last-child{font-weight:bold}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child{font-weight:unset}.mw-parser-output table.numblk>tbody>tr>td:last-child::before{content:"("}.mw-parser-output table.numblk>tbody>tr>td:last-child::after{content:")"}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::before,.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::after{content:none}.mw-parser-output table.numblk>tbody>tr>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:none;border-right:none;border-bottom:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:thin solid;border-right:thin solid;border-bottom:thin solid}.mw-parser-output table.numblk:target{color:var(--color-base,#202122);background-color:#cfe8fd}@media screen{html.skin-theme-clientpref-night .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}</style><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A')={\frac {365}{365}}\times {\frac {364}{365}}\times {\frac {363}{365}}\times {\frac {362}{365}}\times \cdots \times {\frac {343}{365}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>365</mn> <mn>365</mn> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>364</mn> <mn>365</mn> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>363</mn> <mn>365</mn> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>362</mn> <mn>365</mn> </mfrac> </mrow> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>343</mn> <mn>365</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(A')={\frac {365}{365}}\times {\frac {364}{365}}\times {\frac {363}{365}}\times {\frac {362}{365}}\times \cdots \times {\frac {343}{365}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61ae67e88c81cf0e5a104ad945899c61f4ab9fe5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:47.624ex; height:5.176ex;" alt="{\displaystyle P(A')={\frac {365}{365}}\times {\frac {364}{365}}\times {\frac {363}{365}}\times {\frac {362}{365}}\times \cdots \times {\frac {343}{365}}}" /></td> <td></td> <td class="nowrap">1</td></tr></tbody></table> The terms of equation (<a href="#math_1">1</a>) can be collected to arrive at: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A')=\left({\frac {1}{365}}\right)^{23}\times (365\times 364\times 363\times \cdots \times 343)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>365</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msup> <mo>×</mo> <mo stretchy="false">(</mo> <mn>365</mn> <mo>×</mo> <mn>364</mn> <mo>×</mo> <mn>363</mn> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <mn>343</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(A')=\left({\frac {1}{365}}\right)^{23}\times (365\times 364\times 363\times \cdots \times 343)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ede5fde44fedfe898408c1e2e8192d903eb5c423" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.386ex; height:6.509ex;" alt="{\displaystyle P(A')=\left({\frac {1}{365}}\right)^{23}\times (365\times 364\times 363\times \cdots \times 343)}" /></td> <td></td> <td class="nowrap">2</td></tr></tbody></table> Evaluating equation (<a href="#math_2">2</a>) gives P(A′) ≈ 0.492703 Therefore, P(B) ≈ 1 − 0.492703 = 0.507297 (50.7297%). This process can be generalized to a group of n people, where p(n) is the probability of at least two of the n people sharing a birthday. It is easier to first calculate the probability p(n) that all n birthdays are different. According to the <a href="/wiki/Pigeonhole_principle" title="Pigeonhole principle">pigeonhole principle</a>, p(n) is zero when n > 365. When n ≤ 365: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\bar {p}}(n)&=1\times \left(1-{\frac {1}{365}}\right)\times \left(1-{\frac {2}{365}}\right)\times \cdots \times \left(1-{\frac {n-1}{365}}\right)\\[6pt]&={\frac {365\times 364\times \cdots \times (365-n+1)}{365^{n}}}\\[6pt]&={\frac {365!}{365^{n}(365-n)!}}={\frac {n!\cdot {\binom {365}{n}}}{365^{n}}}={\frac {_{365}P_{n}}{365^{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="0.9em 0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>365</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>365</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>365</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>365</mn> <mo>×</mo> <mn>364</mn> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mn>365</mn> <mo>−</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <msup> <mn>365</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>365</mn> <mo>!</mo> </mrow> <mrow> <msup> <mn>365</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>365</mn> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>!</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>365</mn> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mrow> <msup> <mn>365</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>365</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <msup> <mn>365</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\bar {p}}(n)&=1\times \left(1-{\frac {1}{365}}\right)\times \left(1-{\frac {2}{365}}\right)\times \cdots \times \left(1-{\frac {n-1}{365}}\right)\\[6pt]&={\frac {365\times 364\times \cdots \times (365-n+1)}{365^{n}}}\\[6pt]&={\frac {365!}{365^{n}(365-n)!}}={\frac {n!\cdot {\binom {365}{n}}}{365^{n}}}={\frac {_{365}P_{n}}{365^{n}}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97247979ca9c9ce1de6ae233eb2ad1f1bca4ea22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.505ex; width:60.813ex; height:22.176ex;" alt="{\displaystyle {\begin{aligned}{\bar {p}}(n)&=1\times \left(1-{\frac {1}{365}}\right)\times \left(1-{\frac {2}{365}}\right)\times \cdots \times \left(1-{\frac {n-1}{365}}\right)\\[6pt]&={\frac {365\times 364\times \cdots \times (365-n+1)}{365^{n}}}\\[6pt]&={\frac {365!}{365^{n}(365-n)!}}={\frac {n!\cdot {\binom {365}{n}}}{365^{n}}}={\frac {_{365}P_{n}}{365^{n}}}\end{aligned}}}" /></dd></dl> where ! is the <a href="/wiki/Factorial" title="Factorial">factorial</a> operator, (365 n) is the <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficient</a> and kPr denotes <a href="/wiki/Permutation" title="Permutation">permutation</a>. The equation expresses the fact that the first person has no one to share a birthday, the second person cannot have the same birthday as the first (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠364/365⁠), the third cannot have the same birthday as either of the first two (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠363/365⁠), and in general the nth birthday cannot be the same as any of the n − 1 preceding birthdays. The <a href="/wiki/Event_(probability_theory)" title="Event (probability theory)">event</a> of at least two of the n persons having the same birthday is <a href="/wiki/Complementary_event" title="Complementary event">complementary</a> to all n birthdays being different. Therefore, its probability p(n) is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(n)=1-{\bar {p}}(n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(n)=1-{\bar {p}}(n).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c780c6ec48309bcdc9df716561ea7b4083bd25c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:16.774ex; height:2.843ex;" alt="{\displaystyle p(n)=1-{\bar {p}}(n).}" /></dd></dl> The following table shows the probability for some other values of n (for this table, the existence of leap years is ignored, and each birthday is assumed to be equally likely): <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Birthdaymatch.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Birthdaymatch.svg/330px-Birthdaymatch.svg.png" decoding="async" width="310" height="233" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Birthdaymatch.svg/500px-Birthdaymatch.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/06/Birthdaymatch.svg/620px-Birthdaymatch.svg.png 2x" data-file-width="720" data-file-height="540" /></a><figcaption>The probability that no two people share a birthday in a group of n people. Note that the vertical scale is logarithmic (each step down is 1020 times less likely).</figcaption></figure> <dl><dd><table class="wikitable"> <tbody><tr> <th>n</th> <th>p(n) </th></tr> <tr> <td align="right">1</td> <td>00.0% </td></tr> <tr> <td align="right">5</td> <td>02.7% </td></tr> <tr> <td align="right">10</td> <td>11.7% </td></tr> <tr> <td align="right">20</td> <td>41.1% </td></tr> <tr> <td align="right">23</td> <td>50.7% </td></tr> <tr> <td align="right">30</td> <td>70.6% </td></tr> <tr> <td align="right">40</td> <td>89.1% </td></tr> <tr> <td align="right">50</td> <td>97.0% </td></tr> <tr> <td align="right">60</td> <td>99.4% </td></tr> <tr> <td align="right">70</td> <td>99.9% </td></tr> <tr> <td align="right">75</td> <td>99.97% </td></tr> <tr> <td align="right">100</td> <td>99.99997% </td></tr> <tr> <td align="right">200</td> <td>99.9999999999999999999999999998% </td></tr> <tr> <td align="right">300</td> <td>(100 − 6×10−80)% </td></tr> <tr> <td align="right">350</td> <td>(100 − 3×10−129)% </td></tr> <tr> <td align="right">365</td> <td>(100 − 1.45×10−155)% </td></tr> <tr> <td align="right">≥ 366</td> <td>100% </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Approximations">Approximations</h2>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=2" title="Edit section: Approximations">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Birthday_paradox_probability.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Birthday_paradox_probability.svg/310px-Birthday_paradox_probability.svg.png" decoding="async" width="310" height="233" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Birthday_paradox_probability.svg/465px-Birthday_paradox_probability.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Birthday_paradox_probability.svg/620px-Birthday_paradox_probability.svg.png 2x" data-file-width="720" data-file-height="540" /></a><figcaption>Graphs showing the approximate probabilities of at least two people sharing a birthday (<style data-mw-deduplicate="TemplateStyles:r1239334494">@media screen{html.skin-theme-clientpref-night .mw-parser-output div:not(.notheme)>.tmp-color,html.skin-theme-clientpref-night .mw-parser-output p>.tmp-color,html.skin-theme-clientpref-night .mw-parser-output table:not(.notheme) .tmp-color{color:inherit!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output div:not(.notheme)>.tmp-color,html.skin-theme-clientpref-os .mw-parser-output p>.tmp-color,html.skin-theme-clientpref-os .mw-parser-output table:not(.notheme) .tmp-color{color:inherit!important}}</style>red) and its complementary event (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494" />blue)</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Birthday_paradox_approximation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Birthday_paradox_approximation.svg/310px-Birthday_paradox_approximation.svg.png" decoding="async" width="310" height="194" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Birthday_paradox_approximation.svg/465px-Birthday_paradox_approximation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Birthday_paradox_approximation.svg/620px-Birthday_paradox_approximation.svg.png 2x" data-file-width="720" data-file-height="450" /></a><figcaption>A graph showing the accuracy of the approximation 1 − e−n2/730 (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494" />red)</figcaption></figure> The <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> expansion of the <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a> (the constant e ≈ 2.718281828) <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x}=1+x+{\frac {x^{2}}{2!}}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{x}=1+x+{\frac {x^{2}}{2!}}+\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f67772264ca200424ea0f2f4270c7ecb31fba0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.311ex; height:5.843ex;" alt="{\displaystyle e^{x}=1+x+{\frac {x^{2}}{2!}}+\cdots }" /></dd></dl> provides a first-order approximation for ex for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x|\ll 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≪</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|\ll 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a561c2bcbb4c28e95f1ae14e0d11bfd9e2defea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.4ex; height:2.843ex;" alt="{\displaystyle |x|\ll 1}" />: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x}\approx 1+x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>≈</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{x}\approx 1+x.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92fda2fe0341e81c735ec8f237647f4c5dd7e918" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.334ex; height:2.509ex;" alt="{\displaystyle e^{x}\approx 1+x.}" /></dd></dl> To apply this approximation to the first expression derived for p(n), set x = −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠a/365⁠. Thus, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-a/365}\approx 1-{\frac {a}{365}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>365</mn> </mrow> </msup> <mo>≈</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mn>365</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-a/365}\approx 1-{\frac {a}{365}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9e0ea74e03d760c748a9d866edf8433e5e784ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.823ex; height:4.676ex;" alt="{\displaystyle e^{-a/365}\approx 1-{\frac {a}{365}}.}" /></dd></dl> Then, replace a with non-negative integers for each term in the formula of p(n) until a = n − 1, for example, when a = 1, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-1/365}\approx 1-{\frac {1}{365}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>365</mn> </mrow> </msup> <mo>≈</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>365</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-1/365}\approx 1-{\frac {1}{365}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ea599c104d81192e4a969ae87ebe9652452a1db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.776ex; height:5.176ex;" alt="{\displaystyle e^{-1/365}\approx 1-{\frac {1}{365}}.}" /></dd></dl> The first expression derived for p(n) can be approximated as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\bar {p}}(n)&\approx 1\cdot e^{-1/365}\cdot e^{-2/365}\cdots e^{-(n-1)/365}\\[6pt]&=e^{-{\big (}1+2+\,\cdots \,+(n-1){\big )}/365}\\[6pt]&=e^{-{\frac {n(n-1)/2}{365}}}=e^{-{\frac {n(n-1)}{730}}}.\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="0.9em 0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>≈</mo> <mn>1</mn> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>365</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>365</mn> </mrow> </msup> <mo>⋯</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>365</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mspace width="thinmathspace"></mspace> <mo>⋯</mo> <mspace width="thinmathspace"></mspace> <mo>+</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>365</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> <mn>365</mn> </mfrac> </mrow> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>730</mn> </mfrac> </mrow> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\bar {p}}(n)&\approx 1\cdot e^{-1/365}\cdot e^{-2/365}\cdots e^{-(n-1)/365}\\[6pt]&=e^{-{\big (}1+2+\,\cdots \,+(n-1){\big )}/365}\\[6pt]&=e^{-{\frac {n(n-1)/2}{365}}}=e^{-{\frac {n(n-1)}{730}}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa500cfaaebf468593f2fda0474696de149c5ec6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.005ex; width:40.088ex; height:15.176ex;" alt="{\displaystyle {\begin{aligned}{\bar {p}}(n)&\approx 1\cdot e^{-1/365}\cdot e^{-2/365}\cdots e^{-(n-1)/365}\\[6pt]&=e^{-{\big (}1+2+\,\cdots \,+(n-1){\big )}/365}\\[6pt]&=e^{-{\frac {n(n-1)/2}{365}}}=e^{-{\frac {n(n-1)}{730}}}.\end{aligned}}}" /></dd></dl> Therefore, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(n)=1-{\bar {p}}(n)\approx 1-e^{-{\frac {n(n-1)}{730}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>730</mn> </mfrac> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(n)=1-{\bar {p}}(n)\approx 1-e^{-{\frac {n(n-1)}{730}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c2034c62a15b5242af1f417813646cc3bd5fcb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:31.652ex; height:4.509ex;" alt="{\displaystyle p(n)=1-{\bar {p}}(n)\approx 1-e^{-{\frac {n(n-1)}{730}}}.}" /></dd></dl> An even coarser approximation is given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(n)\approx 1-e^{-{\frac {n^{2}}{730}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>730</mn> </mfrac> </mrow> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(n)\approx 1-e^{-{\frac {n^{2}}{730}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e55868abdecc541d6bb9befefa3b711cd9009d13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:17.643ex; height:4.509ex;" alt="{\displaystyle p(n)\approx 1-e^{-{\frac {n^{2}}{730}}},}" /></dd></dl> which, as the graph illustrates, is still fairly accurate. According to the approximation, the same approach can be applied to any number of "people" and "days". If rather than 365 days there are d, if there are n persons, and if n ≪ d, then using the same approach as above we achieve the result that if p(n, d) is the probability that at least two out of n people share the same birthday from a set of d available days, then: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}p(n,d)&\approx 1-e^{-{\frac {n(n-1)}{2d}}}\\[6pt]&\approx 1-e^{-{\frac {n^{2}}{2d}}}.\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="0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>p</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>≈</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>d</mi> </mrow> </mfrac> </mrow> </mrow> </msup> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>≈</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>d</mi> </mrow> </mfrac> </mrow> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p(n,d)&\approx 1-e^{-{\frac {n(n-1)}{2d}}}\\[6pt]&\approx 1-e^{-{\frac {n^{2}}{2d}}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ff46256637894d0250baba3169b2dffd07abf2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:22.252ex; height:10.509ex;" alt="{\displaystyle {\begin{aligned}p(n,d)&\approx 1-e^{-{\frac {n(n-1)}{2d}}}\\[6pt]&\approx 1-e^{-{\frac {n^{2}}{2d}}}.\end{aligned}}}" /></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Simple_exponentiation">Simple exponentiation</h3>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=3" title="Edit section: Simple exponentiation">edit</a>]</div> The probability of any two people not having the same birthday is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠364/365⁠. In a room containing n people, there are (n 2) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠n(n − 1)/2⁠ pairs of people, i.e. (n 2) events. The probability of no two people sharing the same birthday can be approximated by assuming that these events are independent and hence by multiplying their probability together. Being independent would be equivalent to picking <a href="/wiki/Sampling_(statistics)#Replacement_of_selected_units" title="Sampling (statistics)">with replacement</a>, any pair of people in the world, not just in a room. In short <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠364/365⁠ can be multiplied by itself (n 2) times, which gives us <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {p}}(n)\approx \left({\frac {364}{365}}\right)^{\binom {n}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>364</mn> <mn>365</mn> </mfrac> </mrow> <mo>)</mo> </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> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {p}}(n)\approx \left({\frac {364}{365}}\right)^{\binom {n}{2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10c16ab9f10ff711c817eda177cdc0b73416ea39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.089ex; width:19.138ex; height:7.343ex;" alt="{\displaystyle {\bar {p}}(n)\approx \left({\frac {364}{365}}\right)^{\binom {n}{2}}.}" /></dd></dl> Since this is the probability of no one having the same birthday, then the probability of someone sharing a birthday is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(n)\approx 1-\left({\frac {364}{365}}\right)^{\binom {n}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>364</mn> <mn>365</mn> </mfrac> </mrow> <mo>)</mo> </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> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(n)\approx 1-\left({\frac {364}{365}}\right)^{\binom {n}{2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/694bdcfe09974a50f282fa91772fdb18e2d3601f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.089ex; width:22.951ex; height:7.343ex;" alt="{\displaystyle p(n)\approx 1-\left({\frac {364}{365}}\right)^{\binom {n}{2}}.}" /></dd></dl> And for the group of 23 people, the probability of sharing is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(23)\approx 1-\left({\frac {364}{365}}\right)^{\binom {23}{2}}=1-\left({\frac {364}{365}}\right)^{253}\approx 0.500477.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mn>23</mn> <mo stretchy="false">)</mo> <mo>≈</mo> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>364</mn> <mn>365</mn> </mfrac> </mrow> <mo>)</mo> </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>23</mn> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>364</mn> <mn>365</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>253</mn> </mrow> </msup> <mo>≈</mo> <mn>0.500477.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(23)\approx 1-\left({\frac {364}{365}}\right)^{\binom {23}{2}}=1-\left({\frac {364}{365}}\right)^{253}\approx 0.500477.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1a247e3c8098349b29fc1cdc87e869219dd008b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.089ex; width:53.842ex; height:7.343ex;" alt="{\displaystyle p(23)\approx 1-\left({\frac {364}{365}}\right)^{\binom {23}{2}}=1-\left({\frac {364}{365}}\right)^{253}\approx 0.500477.}" /></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Poisson_approximation">Poisson approximation</h3>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=4" title="Edit section: Poisson approximation">edit</a>]</div> Applying the <a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson</a> approximation for the binomial on the group of 23 people, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Poi} \left({\frac {\binom {23}{2}}{365}}\right)=\operatorname {Poi} \left({\frac {253}{365}}\right)\approx \operatorname {Poi} (0.6932)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Poi</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>23</mn> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mn>365</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>Poi</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>253</mn> <mn>365</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mi>Poi</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0.6932</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Poi} \left({\frac {\binom {23}{2}}{365}}\right)=\operatorname {Poi} \left({\frac {253}{365}}\right)\approx \operatorname {Poi} (0.6932)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/727815b043aa343e2c8af1e1ae11627900738910" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:40.678ex; height:7.509ex;" alt="{\displaystyle \operatorname {Poi} \left({\frac {\binom {23}{2}}{365}}\right)=\operatorname {Poi} \left({\frac {253}{365}}\right)\approx \operatorname {Poi} (0.6932)}" /></dd></dl> so <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(X>0)=1-\Pr(X=0)\approx 1-e^{-0.6932}\approx 1-0.499998=0.500002.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>></mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>≈</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>0.6932</mn> </mrow> </msup> <mo>≈</mo> <mn>1</mn> <mo>−</mo> <mn>0.499998</mn> <mo>=</mo> <mn>0.500002.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(X>0)=1-\Pr(X=0)\approx 1-e^{-0.6932}\approx 1-0.499998=0.500002.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b88a049729c8a4efa29920a4287831452ff80f26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:70.868ex; height:3.176ex;" alt="{\displaystyle \Pr(X>0)=1-\Pr(X=0)\approx 1-e^{-0.6932}\approx 1-0.499998=0.500002.}" /></dd></dl> The result is over 50% as previous descriptions. This approximation is the same as the one above based on the Taylor expansion that uses ex ≈ 1 + x. <div class="mw-heading mw-heading3"><h3 id="Square_approximation">Square approximation</h3>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=5" title="Edit section: Square approximation">edit</a>]</div> A good <a href="/wiki/Rule_of_thumb" title="Rule of thumb">rule of thumb</a> which can be used for <a href="/wiki/Mental_calculation" title="Mental calculation">mental calculation</a> is the relation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(n,d)\approx {\frac {n^{2}}{2d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>d</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(n,d)\approx {\frac {n^{2}}{2d}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47df39613cddffa3d9aeed529ac95c1138fdb43a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-left: -0.089ex; width:13.096ex; height:5.843ex;" alt="{\displaystyle p(n,d)\approx {\frac {n^{2}}{2d}}}" /></dd></dl> which can also be written as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\approx {\sqrt {2d\times p(n)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>d</mi> <mo>×</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\approx {\sqrt {2d\times p(n)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18b254f490f0810f3d5305b4434a012686f116e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.409ex; height:4.843ex;" alt="{\displaystyle n\approx {\sqrt {2d\times p(n)}}}" /></dd></dl> which works well for probabilities less than or equal to <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠1/2⁠. In these equations, d is the number of days in a year. For instance, to estimate the number of people required for a <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠1/2⁠ chance of a shared birthday, we get <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\approx {\sqrt {2\times 365\times {\tfrac {1}{2}}}}={\sqrt {365}}\approx 19}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo>×</mo> <mn>365</mn> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>365</mn> </msqrt> </mrow> <mo>≈</mo> <mn>19</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\approx {\sqrt {2\times 365\times {\tfrac {1}{2}}}}={\sqrt {365}}\approx 19}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1e9b97252f9b6fa50a6f40004bdf19ffd21d51b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.751ex; height:4.843ex;" alt="{\displaystyle n\approx {\sqrt {2\times 365\times {\tfrac {1}{2}}}}={\sqrt {365}}\approx 19}" /></dd></dl> Which is not too far from the correct answer of 23. <div class="mw-heading mw-heading3"><h3 id="Approximation_of_number_of_people">Approximation of number of people</h3>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=6" title="Edit section: Approximation of number of people">edit</a>]</div> This can also be approximated using the following formula for the number of people necessary to have at least a <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠1/2⁠ chance of matching: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq {\tfrac {1}{2}}+{\sqrt {{\tfrac {1}{4}}+2\times \ln(2)\times 365}}=22.999943.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mn>2</mn> <mo>×</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mn>365</mn> </msqrt> </mrow> <mo>=</mo> <mn>22.999943.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq {\tfrac {1}{2}}+{\sqrt {{\tfrac {1}{4}}+2\times \ln(2)\times 365}}=22.999943.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be3c27373d0f2fa3bc41a5805613c71c6ff28604" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.747ex; height:4.843ex;" alt="{\displaystyle n\geq {\tfrac {1}{2}}+{\sqrt {{\tfrac {1}{4}}+2\times \ln(2)\times 365}}=22.999943.}" /></dd></dl> This is a result of the good approximation that an event with <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠1/k⁠ probability will have a <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠1/2⁠ chance of occurring at least once if it is repeated k <a href="/wiki/Natural_logarithm_of_2" title="Natural logarithm of 2">ln 2</a> times.<a href="#cite_note-8">[8]</a> <div class="mw-heading mw-heading3"><h3 id="Probability_table">Probability table</h3>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=7" title="Edit section: Probability table">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Birthday_attack" title="Birthday attack">Birthday attack</a></div> <dl><dd><table class="wikitable" style="white-space:nowrap;"> <tbody><tr> <th rowspan="2">length of hex string </th> <th rowspan="2">no. of bits (b) </th> <th rowspan="2">hash space size (2b) </th> <th colspan="10">Number of hashed elements such that probability of at least one hash collision ≥ p </th></tr> <tr> <th>p = 10−18 </th> <th>p = 10−15 </th> <th>p = 10−12 </th> <th>p = 10−9 </th> <th>p = 10−6 </th> <th>p = 0.001 </th> <th>p = 0.01 </th> <th>p = 0.25 </th> <th>p = 0.50 </th> <th>p = 0.75 </th></tr> <tr align="center"> <td bgcolor="#F2F2F2">8 </td> <td bgcolor="#F2F2F2">32 </td> <td bgcolor="#F2F2F2">4.3×109 </td> <td>2 </td> <td>2 </td> <td>2 </td> <td>2.9 </td> <td>93 </td> <td>2.9×103 </td> <td>9.3×103 </td> <td>5.0×104 </td> <td>7.7×104 </td> <td>1.1×105 </td></tr> <tr align="center"> <td bgcolor="#F2F2F2">(10) </td> <td bgcolor="#F2F2F2">(40) </td> <td bgcolor="#F2F2F2">(1.1×1012) </td> <td>2 </td> <td>2 </td> <td>2 </td> <td>47 </td> <td>1.5×103 </td> <td>4.7×104 </td> <td>1.5×105 </td> <td>8.0×105 </td> <td>1.2×106 </td> <td>1.7×106 </td></tr> <tr align="center"> <td bgcolor="#F2F2F2">(12) </td> <td bgcolor="#F2F2F2">(48) </td> <td bgcolor="#F2F2F2">(2.8×1014) </td> <td>2 </td> <td>2 </td> <td>24 </td> <td>7.5×102 </td> <td>2.4×104 </td> <td>7.5×105 </td> <td>2.4×106 </td> <td>1.3×107 </td> <td>2.0×107 </td> <td>2.8×107 </td></tr> <tr align="center"> <td bgcolor="#F2F2F2">16 </td> <td bgcolor="#F2F2F2">64 </td> <td bgcolor="#F2F2F2">1.8×1019 </td> <td>6.1 </td> <td>1.9×102 </td> <td>6.1×103 </td> <td>1.9×105 </td> <td>6.1×106 </td> <td>1.9×108 </td> <td>6.1×108 </td> <td>3.3×109 </td> <td>5.1×109 </td> <td>7.2×109 </td></tr> <tr align="center"> <td bgcolor="#F2F2F2">(24) </td> <td bgcolor="#F2F2F2">(96) </td> <td bgcolor="#F2F2F2">(7.9×1028) </td> <td>4.0×105 </td> <td>1.3×107 </td> <td>4.0×108 </td> <td>1.3×1010 </td> <td>4.0×1011 </td> <td>1.3×1013 </td> <td>4.0×1013 </td> <td>2.1×1014 </td> <td>3.3×1014 </td> <td>4.7×1014 </td></tr> <tr align="center"> <td bgcolor="#F2F2F2">32 </td> <td bgcolor="#F2F2F2">128 </td> <td bgcolor="#F2F2F2">3.4×1038 </td> <td>2.6×1010 </td> <td>8.2×1011 </td> <td>2.6×1013 </td> <td>8.2×1014 </td> <td>2.6×1016 </td> <td>8.3×1017 </td> <td>2.6×1018 </td> <td>1.4×1019 </td> <td>2.2×1019 </td> <td>3.1×1019 </td></tr> <tr align="center"> <td bgcolor="#F2F2F2">(48) </td> <td bgcolor="#F2F2F2">(192) </td> <td bgcolor="#F2F2F2">(6.3×1057) </td> <td>1.1×1020 </td> <td>3.5×1021 </td> <td>1.1×1023 </td> <td>3.5×1024 </td> <td>1.1×1026 </td> <td>3.5×1027 </td> <td>1.1×1028 </td> <td>6.0×1028 </td> <td>9.3×1028 </td> <td>1.3×1029 </td></tr> <tr align="center"> <td bgcolor="#F2F2F2">64 </td> <td bgcolor="#F2F2F2">256 </td> <td bgcolor="#F2F2F2">1.2×1077 </td> <td>4.8×1029 </td> <td>1.5×1031 </td> <td>4.8×1032 </td> <td>1.5×1034 </td> <td>4.8×1035 </td> <td>1.5×1037 </td> <td>4.8×1037 </td> <td>2.6×1038 </td> <td>4.0×1038 </td> <td>5.7×1038 </td></tr> <tr align="center"> <td bgcolor="#F2F2F2">(96) </td> <td bgcolor="#F2F2F2">(384) </td> <td bgcolor="#F2F2F2">(3.9×10115) </td> <td>8.9×1048 </td> <td>2.8×1050 </td> <td>8.9×1051 </td> <td>2.8×1053 </td> <td>8.9×1054 </td> <td>2.8×1056 </td> <td>8.9×1056 </td> <td>4.8×1057 </td> <td>7.4×1057 </td> <td>1.0×1058 </td></tr> <tr align="center"> <td bgcolor="#F2F2F2">128 </td> <td bgcolor="#F2F2F2">512 </td> <td bgcolor="#F2F2F2">1.3×10154 </td> <td>1.6×1068 </td> <td>5.2×1069 </td> <td>1.6×1071 </td> <td>5.2×1072 </td> <td>1.6×1074 </td> <td>5.2×1075 </td> <td>1.6×1076 </td> <td>8.8×1076 </td> <td>1.4×1077 </td> <td>1.9×1077 </td></tr></tbody></table></dd></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Birthday_attack_vs_paradox.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Birthday_attack_vs_paradox.svg/220px-Birthday_attack_vs_paradox.svg.png" decoding="async" width="220" height="367" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Birthday_attack_vs_paradox.svg/330px-Birthday_attack_vs_paradox.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Birthday_attack_vs_paradox.svg/440px-Birthday_attack_vs_paradox.svg.png 2x" data-file-width="512" data-file-height="853" /></a><figcaption>Comparison of the birthday problem (1) and birthday attack (2):<div class="paragraphbreak" style="margin-top:0.5em"></div> In (1), collisions are found within one set, in this case, 3 out of 276 pairings of the 24 lunar astronauts.<div class="paragraphbreak" style="margin-top:0.5em"></div> In (2), collisions are found between two sets, in this case, 1 out of 256 pairings of only the first bytes of SHA-256 hashes of 16 variants each of benign and harmful contracts.</figcaption></figure> The lighter fields in this table show the number of hashes needed to achieve the given probability of collision (column) given a hash space of a certain size in bits (row). Using the birthday analogy: the "hash space size" resembles the "available days", the "probability of collision" resembles the "probability of shared birthday", and the "required number of hashed elements" resembles the "required number of people in a group". One could also use this chart to determine the minimum hash size required (given upper bounds on the hashes and probability of error), or the probability of collision (for fixed number of hashes and probability of error). For comparison, 10−18 to 10−15 is the uncorrectable bit error rate of a typical hard disk.<a href="#cite_note-9">[9]</a> In theory, 128-bit hash functions, such as <a href="/wiki/MD5" title="MD5">MD5</a>, should stay within that range until about 8.2×1011 documents, even if its possible outputs are many more. <div class="mw-heading mw-heading2"><h2 id="An_upper_bound_on_the_probability_and_a_lower_bound_on_the_number_of_people">An upper bound on the probability and a lower bound on the number of people</h2>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=8" title="Edit section: An upper bound on the probability and a lower bound on the number of people">edit</a>]</div> The argument below is adapted from an argument of <a href="/wiki/Paul_Halmos" title="Paul Halmos">Paul Halmos</a>.<a href="#cite_note-10">[nb 1]</a> As stated above, the probability that no two birthdays coincide is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-p(n)={\bar {p}}(n)=\prod _{k=1}^{n-1}\left(1-{\frac {k}{365}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mn>365</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-p(n)={\bar {p}}(n)=\prod _{k=1}^{n-1}\left(1-{\frac {k}{365}}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dae6c92f2ac4124d88bf66b084960db63fb74b37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.392ex; height:7.343ex;" alt="{\displaystyle 1-p(n)={\bar {p}}(n)=\prod _{k=1}^{n-1}\left(1-{\frac {k}{365}}\right).}" /></dd></dl> As in earlier paragraphs, interest lies in the smallest n such that p(n) > <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠1/2⁠; or equivalently, the smallest n such that p(n) < <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠1/2⁠. Using the inequality 1 − x < e−x in the above expression we replace 1 − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠k/365⁠ with e<style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>−k⁄365. This yields <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {p}}(n)=\prod _{k=1}^{n-1}\left(1-{\frac {k}{365}}\right)<\prod _{k=1}^{n-1}\left(e^{-{\frac {k}{365}}}\right)=e^{-{\frac {n(n-1)}{730}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mn>365</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo><</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mn>365</mn> </mfrac> </mrow> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>730</mn> </mfrac> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {p}}(n)=\prod _{k=1}^{n-1}\left(1-{\frac {k}{365}}\right)<\prod _{k=1}^{n-1}\left(e^{-{\frac {k}{365}}}\right)=e^{-{\frac {n(n-1)}{730}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34c4bb506ff584ced511c4c281ff742302c2946b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.089ex; width:49.92ex; height:7.343ex;" alt="{\displaystyle {\bar {p}}(n)=\prod _{k=1}^{n-1}\left(1-{\frac {k}{365}}\right)<\prod _{k=1}^{n-1}\left(e^{-{\frac {k}{365}}}\right)=e^{-{\frac {n(n-1)}{730}}}.}" /></dd></dl> Therefore, the expression above is not only an approximation, but also an <a href="/wiki/Upper_bound" class="mw-redirect" title="Upper bound">upper bound</a> of p(n). The inequality <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-{\frac {n(n-1)}{730}}}<{\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>730</mn> </mfrac> </mrow> </mrow> </msup> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-{\frac {n(n-1)}{730}}}<{\frac {1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c497cc849d78859fb7ddcdd47a56708e83b4d8b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.873ex; height:5.509ex;" alt="{\displaystyle e^{-{\frac {n(n-1)}{730}}}<{\frac {1}{2}}}" /></dd></dl> implies p(n) < <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠1/2⁠. Solving for n gives <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{2}-n>730\ln 2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>n</mi> <mo>></mo> <mn>730</mn> <mi>ln</mi> <mo>⁡</mo> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{2}-n>730\ln 2.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f993e1a7d3f9f66b55169f0b5f9d714f4ca05b99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.793ex; height:2.843ex;" alt="{\displaystyle n^{2}-n>730\ln 2.}" /></dd></dl> Now, 730 ln 2 is approximately 505.997, which is barely below 506, the value of n2 − n attained when n = 23. Therefore, 23 people suffice. Incidentally, solving n2 − n = 730 ln 2 for n gives the approximate formula of Frank H. Mathis cited above. This derivation only shows that at most 23 people are needed to ensure the chances of a birthday match are at least even; it leaves open the possibility that n is 22 or less could also work. <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=9" title="Edit section: Generalizations">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Arbitrary_number_of_days">Arbitrary number of days</h3>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=10" title="Edit section: Arbitrary number of days">edit</a>]</div> Given a year with d days, the generalized birthday problem asks for the minimal number n(d) such that, in a set of n randomly chosen people, the probability of a birthday coincidence is at least 50%. In other words, n(d) is the minimal integer n such that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-\left(1-{\frac {1}{d}}\right)\left(1-{\frac {2}{d}}\right)\cdots \left(1-{\frac {n-1}{d}}\right)\geq {\frac {1}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>d</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>d</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>⋯</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>d</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-\left(1-{\frac {1}{d}}\right)\left(1-{\frac {2}{d}}\right)\cdots \left(1-{\frac {n-1}{d}}\right)\geq {\frac {1}{2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37a5f7575c11102f96165a8ad63168722be3c312" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:46.241ex; height:6.176ex;" alt="{\displaystyle 1-\left(1-{\frac {1}{d}}\right)\left(1-{\frac {2}{d}}\right)\cdots \left(1-{\frac {n-1}{d}}\right)\geq {\frac {1}{2}}.}" /></dd></dl> The classical birthday problem thus corresponds to determining n(365). The first 99 values of n(d) are given here (sequence <a href="//oeis.org/A033810" class="extiw" title="oeis:A033810">A033810</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>): <dl><dd><table class="wikitable" style="text-align:center;"> <tbody><tr> <th scope="row">d </th> <td>1–2</td> <td>3–5</td> <td>6–9</td> <td>10–16</td> <td>17–23</td> <td>24–32</td> <td>33–42</td> <td>43–54</td> <td>55–68</td> <td>69–82</td> <td>83–99 </td></tr> <tr> <th scope="row">n(d) </th> <td>2</td> <td>3</td> <td>4</td> <td>5</td> <td>6</td> <td>7</td> <td>8</td> <td>9</td> <td>10</td> <td>11</td> <td>12 </td></tr></tbody></table></dd></dl> A similar calculation shows that n(d) = 23 when d is in the range 341–372. A number of bounds and formulas for n(d) have been published.<a href="#cite_note-11">[10]</a> For any d ≥ 1, the number n(d) satisfies<a href="#cite_note-12">[11]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {3-2\ln 2}{6}}<n(d)-{\sqrt {2d\ln 2}}\leq 9-{\sqrt {86\ln 2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>−</mo> <mn>2</mn> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> </mrow> <mn>6</mn> </mfrac> </mrow> <mo><</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>d</mi> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> </msqrt> </mrow> <mo>≤</mo> <mn>9</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>86</mn> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3-2\ln 2}{6}}<n(d)-{\sqrt {2d\ln 2}}\leq 9-{\sqrt {86\ln 2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a15aa098d02711e1653197be9c6a725f3b08228" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.311ex; height:5.343ex;" alt="{\displaystyle {\frac {3-2\ln 2}{6}}<n(d)-{\sqrt {2d\ln 2}}\leq 9-{\sqrt {86\ln 2}}.}" /></dd></dl> These bounds are optimal in the sense that the sequence n(d) − √2d ln 2 gets arbitrarily close to <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {3-2\ln 2}{6}}\approx 0.27,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>−</mo> <mn>2</mn> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> </mrow> <mn>6</mn> </mfrac> </mrow> <mo>≈</mo> <mn>0.27</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3-2\ln 2}{6}}\approx 0.27,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c037ae19a768e6ef92f7326003e1d5cd7e2d473" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.757ex; height:5.343ex;" alt="{\displaystyle {\frac {3-2\ln 2}{6}}\approx 0.27,}" /></dd></dl> while it has <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 9-{\sqrt {86\ln 2}}\approx 1.28}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>9</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>86</mn> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> </msqrt> </mrow> <mo>≈</mo> <mn>1.28</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 9-{\sqrt {86\ln 2}}\approx 1.28}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed64e26187df414396d62cfaf9509cf3a3c01b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.372ex; height:3.009ex;" alt="{\displaystyle 9-{\sqrt {86\ln 2}}\approx 1.28}" /></dd></dl> as its maximum, taken for d = 43. The bounds are sufficiently tight to give the exact value of n(d) in most of the cases. For example, for d = 365 these bounds imply that 22.7633 < n(365) < 23.7736 and 23 is the only integer in that range. In general, it follows from these bounds that n(d) always equals either <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\lceil {\sqrt {2d\ln 2}}\,\right\rceil \quad {\text{or}}\quad \left\lceil {\sqrt {2d\ln 2}}\,\right\rceil +1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⌈</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>d</mi> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace"></mspace> </mrow> <mo>⌉</mo> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>or</mtext> </mrow> <mspace width="1em"></mspace> <mrow> <mo>⌈</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>d</mi> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace"></mspace> </mrow> <mo>⌉</mo> </mrow> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\lceil {\sqrt {2d\ln 2}}\,\right\rceil \quad {\text{or}}\quad \left\lceil {\sqrt {2d\ln 2}}\,\right\rceil +1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e678971307161ca29be6609e2da96d071f47b63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.041ex; height:3.343ex;" alt="{\displaystyle \left\lceil {\sqrt {2d\ln 2}}\,\right\rceil \quad {\text{or}}\quad \left\lceil {\sqrt {2d\ln 2}}\,\right\rceil +1}" /></dd></dl> where ⌈ · ⌉ denotes the <a href="/wiki/Floor_and_ceiling_functions" title="Floor and ceiling functions">ceiling function</a>. The formula <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n(d)=\left\lceil {\sqrt {2d\ln 2}}\,\right\rceil }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>⌈</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>d</mi> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace"></mspace> </mrow> <mo>⌉</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n(d)=\left\lceil {\sqrt {2d\ln 2}}\,\right\rceil }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a75a0890e5095d825eaf98d3acdbe2bef987e69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.291ex; height:3.343ex;" alt="{\displaystyle n(d)=\left\lceil {\sqrt {2d\ln 2}}\,\right\rceil }" /></dd></dl> holds for 73% of all integers d.<a href="#cite_note-Brink-13">[12]</a> The formula <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n(d)=\left\lceil {\sqrt {2d\ln 2}}+{\frac {3-2\ln 2}{6}}\right\rceil }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>⌈</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>d</mi> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>−</mo> <mn>2</mn> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> </mrow> <mn>6</mn> </mfrac> </mrow> </mrow> <mo>⌉</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n(d)=\left\lceil {\sqrt {2d\ln 2}}+{\frac {3-2\ln 2}{6}}\right\rceil }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bab0013af45547a93ffce40b1239e8bb17c93bdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.137ex; height:6.176ex;" alt="{\displaystyle n(d)=\left\lceil {\sqrt {2d\ln 2}}+{\frac {3-2\ln 2}{6}}\right\rceil }" /></dd></dl> holds for <a href="/wiki/Almost_all" title="Almost all">almost all</a> d, i.e., for a set of integers d with <a href="/wiki/Asymptotic_density" class="mw-redirect" title="Asymptotic density">asymptotic density</a> 1.<a href="#cite_note-Brink-13">[12]</a> The formula <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n(d)=\left\lceil {\sqrt {2d\ln 2}}+{\frac {3-2\ln 2}{6}}+{\frac {9-4(\ln 2)^{2}}{72{\sqrt {2d\ln 2}}}}\right\rceil }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>⌈</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>d</mi> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>−</mo> <mn>2</mn> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> </mrow> <mn>6</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>9</mn> <mo>−</mo> <mn>4</mn> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>72</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>d</mi> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo>⌉</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n(d)=\left\lceil {\sqrt {2d\ln 2}}+{\frac {3-2\ln 2}{6}}+{\frac {9-4(\ln 2)^{2}}{72{\sqrt {2d\ln 2}}}}\right\rceil }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/418edd16de5a67451c10aa2574e9a1c959413d47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:46.591ex; height:7.509ex;" alt="{\displaystyle n(d)=\left\lceil {\sqrt {2d\ln 2}}+{\frac {3-2\ln 2}{6}}+{\frac {9-4(\ln 2)^{2}}{72{\sqrt {2d\ln 2}}}}\right\rceil }" /></dd></dl> holds for all d ≤ 1018, but it is conjectured that there are infinitely many counterexamples to this formula.<a href="#cite_note-ReferenceA-14">[13]</a> The formula <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n(d)=\left\lceil {\sqrt {2d\ln 2}}+{\frac {3-2\ln 2}{6}}+{\frac {9-4(\ln 2)^{2}}{72{\sqrt {2d\ln 2}}}}-{\frac {2(\ln 2)^{2}}{135d}}\right\rceil }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>⌈</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>d</mi> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>−</mo> <mn>2</mn> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> </mrow> <mn>6</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>9</mn> <mo>−</mo> <mn>4</mn> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>72</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>d</mi> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>135</mn> <mi>d</mi> </mrow> </mfrac> </mrow> </mrow> <mo>⌉</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n(d)=\left\lceil {\sqrt {2d\ln 2}}+{\frac {3-2\ln 2}{6}}+{\frac {9-4(\ln 2)^{2}}{72{\sqrt {2d\ln 2}}}}-{\frac {2(\ln 2)^{2}}{135d}}\right\rceil }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f5e9f355d6e76e8a5fc62d85ddc3bb0d7866141" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:57.783ex; height:7.509ex;" alt="{\displaystyle n(d)=\left\lceil {\sqrt {2d\ln 2}}+{\frac {3-2\ln 2}{6}}+{\frac {9-4(\ln 2)^{2}}{72{\sqrt {2d\ln 2}}}}-{\frac {2(\ln 2)^{2}}{135d}}\right\rceil }" /></dd></dl> holds for all d ≤ 1018, and it is conjectured that this formula holds for all d.<a href="#cite_note-ReferenceA-14">[13]</a> <div class="mw-heading mw-heading3"><h3 id="More_than_two_people_sharing_a_birthday">More than two people sharing a birthday</h3>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=11" title="Edit section: More than two people sharing a birthday">edit</a>]</div> It is possible to extend the problem to ask how many people in a group are necessary for there to be a greater than 50% probability that at least 3, 4, 5, etc. of the group share the same birthday. The first few values are as follows: >50% probability of 3 people sharing a birthday - 88 people; >50% probability of 4 people sharing a birthday - 187 people (sequence <a href="//oeis.org/A014088" class="extiw" title="oeis:A014088">A014088</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).<a href="#cite_note-15">[14]</a> <div class="mw-heading mw-heading3"><h3 id="Everyone_shares_a_birthday">Everyone shares a birthday</h3>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=12" title="Edit section: Everyone shares a birthday">edit</a>]</div> The strong birthday problem asks for the number of people that need to be gathered together before there is a 50% chance that everyone in the gathering shares their birthday with at least one other person. For d=365 days the answer is 3,064 people.<a href="#cite_note-16">[15]</a><a href="#cite_note-17">[16]</a> The number of people needed for arbitrary number of days is given by (sequence <a href="//oeis.org/A380129" class="extiw" title="oeis:A380129">A380129</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) <div class="mw-heading mw-heading3"><h3 id="Probability_of_a_shared_birthday_(collision)">Probability of a shared birthday (collision)</h3>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=13" title="Edit section: Probability of a shared birthday (collision)">edit</a>]</div> The birthday problem can be generalized as follows: <dl><dd>Given n random integers drawn from a <a href="/wiki/Uniform_distribution_(discrete)" class="mw-redirect" title="Uniform distribution (discrete)">discrete uniform distribution</a> with range [1,d], what is the probability p(n; d) that at least two numbers are the same? (d = 365 gives the usual birthday problem.)<a href="#cite_note-18">[17]</a></dd></dl> The generic results can be derived using the same arguments given above. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}p(n;d)&={\begin{cases}1-\displaystyle \prod _{k=1}^{n-1}\left(1-{\frac {k}{d}}\right)&n\leq d\\1&n>d\end{cases}}\\[8px]&\approx 1-e^{-{\frac {n(n-1)}{2d}}}\\&\approx 1-\left({\frac {d-1}{d}}\right)^{\frac {n(n-1)}{2}}\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 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>p</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>;</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> <mo>−</mo> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>d</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mtd> <mtd> <mi>n</mi> <mo>≤</mo> <mi>d</mi> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi>n</mi> <mo>></mo> <mi>d</mi> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>≈</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>d</mi> </mrow> </mfrac> </mrow> </mrow> </msup> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>≈</mo> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>d</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p(n;d)&={\begin{cases}1-\displaystyle \prod _{k=1}^{n-1}\left(1-{\frac {k}{d}}\right)&n\leq d\\1&n>d\end{cases}}\\[8px]&\approx 1-e^{-{\frac {n(n-1)}{2d}}}\\&\approx 1-\left({\frac {d-1}{d}}\right)^{\frac {n(n-1)}{2}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85e10a5ef850209f4f2010865e273fc0ce51d35b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.671ex; width:38.275ex; height:24.509ex;" alt="{\displaystyle {\begin{aligned}p(n;d)&={\begin{cases}1-\displaystyle \prod _{k=1}^{n-1}\left(1-{\frac {k}{d}}\right)&n\leq d\\1&n>d\end{cases}}\\[8px]&\approx 1-e^{-{\frac {n(n-1)}{2d}}}\\&\approx 1-\left({\frac {d-1}{d}}\right)^{\frac {n(n-1)}{2}}\end{aligned}}}" /></dd></dl> Conversely, if n(p; d) denotes the number of random integers drawn from [1,d] to obtain a probability p that at least two numbers are the same, then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n(p;d)\approx {\sqrt {2d\cdot \ln \left({\frac {1}{1-p}}\right)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>;</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>d</mi> <mo>⋅</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n(p;d)\approx {\sqrt {2d\cdot \ln \left({\frac {1}{1-p}}\right)}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829452a9865564f9c8918e5316099069dbc913dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.119ex; height:7.509ex;" alt="{\displaystyle n(p;d)\approx {\sqrt {2d\cdot \ln \left({\frac {1}{1-p}}\right)}}.}" /></dd></dl> The birthday problem in this more generic sense applies to <a href="/wiki/Hash_function" title="Hash function">hash functions</a>: the expected number of N-<a href="/wiki/Bit" title="Bit">bit</a> hashes that can be generated before getting a collision is not 2N, but rather only 2<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027" />N⁄2. This is exploited by <a href="/wiki/Birthday_attack" title="Birthday attack">birthday attacks</a> on <a href="/wiki/Cryptographic_hash_function" title="Cryptographic hash function">cryptographic hash functions</a> and is the reason why a small number of collisions in a <a href="/wiki/Hash_table" title="Hash table">hash table</a> are, for all practical purposes, inevitable. The theory behind the birthday problem was used by Zoe Schnabel<a href="#cite_note-19">[18]</a> under the name of <a href="/wiki/Mark_and_recapture" title="Mark and recapture">capture-recapture</a> statistics to estimate the size of fish population in lakes. The birthday problem and its generalizations are also useful tools for modelling coincidences.<a href="#cite_note-Pollanen-20">[19]</a> <div class="mw-heading mw-heading4"><h4 id="Probability_of_a_unique_collision">Probability of a unique collision</h4>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=14" title="Edit section: Probability of a unique collision">edit</a>]</div> The classic birthday problem allows for more than two people to share a particular birthday or for there to be matches on multiple days. The probability that among n people there is exactly one pair of individuals with a matching birthday given d possible days is<a href="#cite_note-Pollanen-20">[19]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{2}(n;d)={\frac {n \choose 2}{d-n+1}}(1-p(n;d))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo>;</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow> <mi>d</mi> <mo>−</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>;</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{2}(n;d)={\frac {n \choose 2}{d-n+1}}(1-p(n;d))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d2d7ac231d3467dbfd708946e0f8e63cec69d2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; margin-left: -0.089ex; width:33.591ex; height:6.343ex;" alt="{\displaystyle p_{2}(n;d)={\frac {n \choose 2}{d-n+1}}(1-p(n;d))}" /></dd></dl> Unlike the standard birthday problem, as n increases the probability reaches a maximum value before decreasing. For example, for d = 365, the probability of a unique match has a maximum value of 0.3864 occurring when n = 28. <div class="mw-heading mw-heading4"><h4 id="Generalization_to_multiple_types_of_people">Generalization to multiple types of people</h4>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=15" title="Edit section: Generalization to multiple types of people">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:2d_birthday.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a1/2d_birthday.png/220px-2d_birthday.png" decoding="async" width="220" height="154" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a1/2d_birthday.png/330px-2d_birthday.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a1/2d_birthday.png/440px-2d_birthday.png 2x" data-file-width="915" data-file-height="642" /></a><figcaption>Plot of the probability of at least one shared birthday between at least one man and one woman</figcaption></figure> The basic problem considers all trials to be of one "type". The birthday problem has been generalized to consider an arbitrary number of types.<a href="#cite_note-21">[20]</a> In the simplest extension there are two types of people, say m men and n women, and the problem becomes characterizing the probability of a shared birthday between at least one man and one woman. (Shared birthdays between two men or two women do not count.) The probability of no shared birthdays here is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{0}={\frac {1}{d^{m+n}}}\sum _{i=1}^{m}\sum _{j=1}^{n}S_{2}(m,i)S_{2}(n,j)\prod _{k=0}^{i+j-1}d-k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mi>n</mi> </mrow> </msup> </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>m</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>i</mi> <mo stretchy="false">)</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mi>j</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>d</mi> <mo>−</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{0}={\frac {1}{d^{m+n}}}\sum _{i=1}^{m}\sum _{j=1}^{n}S_{2}(m,i)S_{2}(n,j)\prod _{k=0}^{i+j-1}d-k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8e89f3aef118c3ea0df824afb46eedda6869acc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; margin-left: -0.089ex; width:45.783ex; height:7.676ex;" alt="{\displaystyle p_{0}={\frac {1}{d^{m+n}}}\sum _{i=1}^{m}\sum _{j=1}^{n}S_{2}(m,i)S_{2}(n,j)\prod _{k=0}^{i+j-1}d-k}" /></dd></dl> where d = 365 and S2 are <a href="/wiki/Stirling_numbers_of_the_second_kind" title="Stirling numbers of the second kind">Stirling numbers of the second kind</a>. Consequently, the desired probability is 1 − p0. This variation of the birthday problem is interesting because there is not a unique solution for the total number of people m + n. For example, the usual 50% probability value is realized for both a 32-member group of 16 men and 16 women and a 49-member group of 43 women and 6 men. <div class="mw-heading mw-heading2"><h2 id="Other_birthday_problems">Other birthday problems</h2>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=16" title="Edit section: Other birthday problems">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="First_match">First match</h3>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=17" title="Edit section: First match">edit</a>]</div> A related question is, as people enter a room one at a time, which one is most likely to be the first to have the same birthday as someone already in the room? That is, for what n is p(n) − p(n − 1) maximum? The answer is 20—if there is a prize for first match, the best position in line is 20th.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <div class="mw-heading mw-heading3"><h3 id="Same_birthday_as_you">Same birthday as you</h3>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=18" title="Edit section: Same birthday as you">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Birthday_paradox.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Birthday_paradox.svg/310px-Birthday_paradox.svg.png" decoding="async" width="310" height="157" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Birthday_paradox.svg/465px-Birthday_paradox.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Birthday_paradox.svg/620px-Birthday_paradox.svg.png 2x" data-file-width="575" data-file-height="291" /></a><figcaption>Comparing p(n) = probability of a birthday match with q(n) = probability of matching your birthday</figcaption></figure> In the birthday problem, neither of the two people is chosen in advance. By contrast, the probability q(n) that at least one other person in a room of n other people has the same birthday as a particular person (for example, you) is given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(n)=1-\left({\frac {365-1}{365}}\right)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>365</mn> <mo>−</mo> <mn>1</mn> </mrow> <mn>365</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q(n)=1-\left({\frac {365-1}{365}}\right)^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/862a88ac6f761e58c4234ebd987d5496dad56104" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.341ex; height:6.176ex;" alt="{\displaystyle q(n)=1-\left({\frac {365-1}{365}}\right)^{n}}" /></dd></dl> and for general d by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(n;d)=1-\left({\frac {d-1}{d}}\right)^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>;</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>d</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q(n;d)=1-\left({\frac {d-1}{d}}\right)^{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9caa862e83c614d6ede9746b71bd0411370c9eab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.966ex; height:6.176ex;" alt="{\displaystyle q(n;d)=1-\left({\frac {d-1}{d}}\right)^{n}.}" /></dd></dl> In the standard case of d = 365, substituting n = 23 gives about 6.1%, which is less than 1 chance in 16. For a greater than 50% chance that at least one other person in a roomful of n people has the same birthday as you, n would need to be at least 253. This number is significantly higher than <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠365/2⁠ = 182.5: the reason is that it is likely that there are some birthday matches among the other people in the room. <div class="mw-heading mw-heading3"><h3 id="Number_of_people_with_a_shared_birthday">Number of people with a shared birthday</h3>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=19" title="Edit section: Number of people with a shared birthday">edit</a>]</div> For any one person in a group of n people the probability that he or she shares his birthday with someone else is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(n-1;d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>;</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q(n-1;d)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8e7eb5f9b1849ba7a61903ebc8ad5dfe3abb650" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.526ex; height:2.843ex;" alt="{\displaystyle q(n-1;d)}" />, as explained above. The expected number of people with a shared (non-unique) birthday can now be calculated easily by multiplying that probability by the number of people (n), so it is: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\left(1-\left({\frac {d-1}{d}}\right)^{n-1}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>d</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\left(1-\left({\frac {d-1}{d}}\right)^{n-1}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09fb6f1ce193a3c6ec5bdb0ec8c110a568873099" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.261ex; height:7.509ex;" alt="{\displaystyle n\left(1-\left({\frac {d-1}{d}}\right)^{n-1}\right)}" /></dd></dl> (This multiplication can be done this way because of the linearity of the <a href="/wiki/Expected_value" title="Expected value">expected value</a> of indicator variables). This implies that the expected number of people with a non-shared (unique) birthday is: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\left({\frac {d-1}{d}}\right)^{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>d</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\left({\frac {d-1}{d}}\right)^{n-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31d77ae808ca08645fd81e73a4ce21bef5a74a6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.19ex; height:6.509ex;" alt="{\displaystyle n\left({\frac {d-1}{d}}\right)^{n-1}}" /></dd></dl> Similar formulas can be derived for the expected number of people who share with three, four, etc. other people. <div class="mw-heading mw-heading3"><h3 id="Number_of_people_until_every_birthday_is_achieved">Number of people until every birthday is achieved</h3>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=20" title="Edit section: Number of people until every birthday is achieved">edit</a>]</div> The expected number of people needed until every birthday is achieved is called the <a href="/wiki/Coupon_collector%27s_problem" title="Coupon collector's problem">Coupon collector's problem</a>. It can be calculated by nHn, where Hn is the nth <a href="/wiki/Harmonic_number" title="Harmonic number">harmonic number</a>. For 365 possible dates (the birthday problem), the answer is 2365. <div class="mw-heading mw-heading3"><h3 id="Near_matches">Near matches</h3>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=21" title="Edit section: Near matches">edit</a>]</div> Another generalization is to ask for the probability of finding at least one pair in a group of n people with birthdays within k calendar days of each other, if there are d equally likely birthdays.<a href="#cite_note-abramson-22">[21]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}p(n,k,d)&=1-{\frac {(d-nk-1)!}{d^{n-1}{\bigl (}d-n(k+1){\bigr )}!}}\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>p</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>−</mo> <mi>n</mi> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>d</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>!</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p(n,k,d)&=1-{\frac {(d-nk-1)!}{d^{n-1}{\bigl (}d-n(k+1){\bigr )}!}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2d5f640eec4e46c0b1ab2560a8e217b3a88a327" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:37.345ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}p(n,k,d)&=1-{\frac {(d-nk-1)!}{d^{n-1}{\bigl (}d-n(k+1){\bigr )}!}}\end{aligned}}}" /></dd></dl> The number of people required so that the probability that some pair will have a birthday separated by k days or fewer will be higher than 50% is given in the following table: <dl><dd><table class="wikitable" style="text-align: center"> <tbody><tr> <th>k</th> <th>n for d = 365 </th></tr> <tr> <td>0</td> <td>23 </td></tr> <tr> <td>1</td> <td>14 </td></tr> <tr> <td>2</td> <td>11 </td></tr> <tr> <td>3</td> <td>9 </td></tr> <tr> <td>4</td> <td>8 </td></tr> <tr> <td>5</td> <td>8 </td></tr> <tr> <td>6</td> <td>7 </td></tr> <tr> <td>7</td> <td>7 </td></tr></tbody></table></dd></dl> Thus in a group of just seven random people, it is more likely than not that two of them will have a birthday within a week of each other.<a href="#cite_note-abramson-22">[21]</a> <div class="mw-heading mw-heading3"><h3 id="Number_of_days_with_a_certain_number_of_birthdays">Number of days with a certain number of birthdays</h3>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=22" title="Edit section: Number of days with a certain number of birthdays">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Number_of_days_with_at_least_one_birthday">Number of days with at least one birthday</h4>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=23" title="Edit section: Number of days with at least one birthday">edit</a>]</div> The expected number of different birthdays, i.e. the number of days that are at least one person's birthday, is: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d-d\left({\frac {d-1}{d}}\right)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>−</mo> <mi>d</mi> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>d</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d-d\left({\frac {d-1}{d}}\right)^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc6b7217a6040a657390a5bc95e7212e6637dcb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.967ex; height:6.176ex;" alt="{\displaystyle d-d\left({\frac {d-1}{d}}\right)^{n}}" /></dd></dl> This follows from the expected number of days that are no one's birthday: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\left({\frac {d-1}{d}}\right)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>d</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\left({\frac {d-1}{d}}\right)^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43e5d69466ab4c228c9375fa6b925870fcc1b338" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.91ex; height:6.176ex;" alt="{\displaystyle d\left({\frac {d-1}{d}}\right)^{n}}" /></dd></dl> which follows from the probability that a particular day is no one's birthday, (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠d − 1/d⁠)n  , easily summed because of the linearity of the expected value. For instance, with <var style="padding-right: 1px;">d</var> = 365, you should expect about 21 different birthdays when there are 22 people, or 46 different birthdays when there are 50 people. When there are 1000 people, there will be around 341 different birthdays (24 unclaimed birthdays). <div class="mw-heading mw-heading4"><h4 id="Number_of_days_with_at_least_two_birthdays">Number of days with at least two birthdays</h4>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=24" title="Edit section: Number of days with at least two birthdays">edit</a>]</div> The above can be generalized from the distribution of the number of people with their birthday on any particular day, which is a <a href="/wiki/Binomial_distribution" title="Binomial distribution">Binomial distribution</a> with probability <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠1/d⁠. Multiplying the relevant probability by d will then give the expected number of days. For example, the expected number of days which are shared; i.e. which are at least two (i.e. not zero and not one) people's birthday is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d-d\left({\frac {d-1}{d}}\right)^{n}-d\cdot {\binom {n}{1}}\left({\frac {1}{d}}\right)^{1}\left({\frac {d-1}{d}}\right)^{n-1}=d-d\left({\frac {d-1}{d}}\right)^{n}-n\left({\frac {d-1}{d}}\right)^{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>−</mo> <mi>d</mi> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>d</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mi>d</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>1</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>d</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>d</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mi>d</mi> <mo>−</mo> <mi>d</mi> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>d</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mi>n</mi> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>d</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d-d\left({\frac {d-1}{d}}\right)^{n}-d\cdot {\binom {n}{1}}\left({\frac {1}{d}}\right)^{1}\left({\frac {d-1}{d}}\right)^{n-1}=d-d\left({\frac {d-1}{d}}\right)^{n}-n\left({\frac {d-1}{d}}\right)^{n-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faf970814a849410522f8922740208e70c07be9b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:81.936ex; height:6.509ex;" alt="{\displaystyle d-d\left({\frac {d-1}{d}}\right)^{n}-d\cdot {\binom {n}{1}}\left({\frac {1}{d}}\right)^{1}\left({\frac {d-1}{d}}\right)^{n-1}=d-d\left({\frac {d-1}{d}}\right)^{n}-n\left({\frac {d-1}{d}}\right)^{n-1}}" /> <div class="mw-heading mw-heading3"><h3 id="Number_of_people_who_repeat_a_birthday">Number of people who repeat a birthday</h3>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=25" title="Edit section: Number of people who repeat a birthday">edit</a>]</div> The probability that the kth integer randomly chosen from [1,d] will repeat at least one previous choice equals q(k − 1; d) above. The expected total number of times a selection will repeat a previous selection as n such integers are chosen equals<a href="#cite_note-23">[22]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{n}q(k-1;d)=n-d+d\left({\frac {d-1}{d}}\right)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>q</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo>;</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo>−</mo> <mi>d</mi> <mo>+</mo> <mi>d</mi> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>d</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{n}q(k-1;d)=n-d+d\left({\frac {d-1}{d}}\right)^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e47bd925574af31a885fb12068ac3f66f953d76f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.385ex; height:6.843ex;" alt="{\displaystyle \sum _{k=1}^{n}q(k-1;d)=n-d+d\left({\frac {d-1}{d}}\right)^{n}}" /></dd></dl> This can be seen to equal the number of people minus the expected number of different birthdays. <div class="mw-heading mw-heading3"><h3 id="Average_number_of_people_to_get_at_least_one_shared_birthday">Average number of people to get at least one shared birthday</h3>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=26" title="Edit section: Average number of people to get at least one shared birthday">edit</a>]</div> In an alternative formulation of the birthday problem, one asks the average number of people required to find a pair with the same birthday. If we consider the probability function Pr[n people have at least one shared birthday], this average is determining the <a href="/wiki/Mean" title="Mean">mean</a> of the distribution, as opposed to the customary formulation, which asks for the <a href="/wiki/Median" title="Median">median</a>. The problem is relevant to several <a href="/wiki/Hash_function" title="Hash function">hashing algorithms</a> analyzed by <a href="/wiki/Donald_Knuth" title="Donald Knuth">Donald Knuth</a> in his book <a href="/wiki/The_Art_of_Computer_Programming" title="The Art of Computer Programming">The Art of Computer Programming</a>. It may be shown<a href="#cite_note-knuth73-24">[23]</a><a href="#cite_note-flajolet95-25">[24]</a> that if one samples uniformly, with replacement, from a population of size M, the number of trials required for the first repeated sampling of some individual has <a href="/wiki/Expected_value" title="Expected value">expected value</a> n = 1 + Q(M), where <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(M)=\sum _{k=1}^{M}{\frac {M!}{(M-k)!M^{k}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>M</mi> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(M)=\sum _{k=1}^{M}{\frac {M!}{(M-k)!M^{k}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dda887b3009cd3f825db80878f23827258bb725" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.951ex; height:7.343ex;" alt="{\displaystyle Q(M)=\sum _{k=1}^{M}{\frac {M!}{(M-k)!M^{k}}}.}" /></dd></dl> The function <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(M)=1+{\frac {M-1}{M}}+{\frac {(M-1)(M-2)}{M^{2}}}+\cdots +{\frac {(M-1)(M-2)\cdots 1}{M^{M-1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>M</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⋯</mo> <mn>1</mn> </mrow> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(M)=1+{\frac {M-1}{M}}+{\frac {(M-1)(M-2)}{M^{2}}}+\cdots +{\frac {(M-1)(M-2)\cdots 1}{M^{M-1}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2ea302bcbe68051335e63771b23947ba0b0eb18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:71.066ex; height:6.009ex;" alt="{\displaystyle Q(M)=1+{\frac {M-1}{M}}+{\frac {(M-1)(M-2)}{M^{2}}}+\cdots +{\frac {(M-1)(M-2)\cdots 1}{M^{M-1}}}}" /></dd></dl> has been studied by <a href="/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan">Srinivasa Ramanujan</a> and has <a href="/wiki/Asymptotic_expansion" title="Asymptotic expansion">asymptotic expansion</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(M)\sim {\sqrt {\frac {\pi M}{2}}}-{\frac {1}{3}}+{\frac {1}{12}}{\sqrt {\frac {\pi }{2M}}}-{\frac {4}{135M}}+\cdots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>π</mi> <mi>M</mi> </mrow> <mn>2</mn> </mfrac> </msqrt> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>π</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> </mfrac> </msqrt> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mrow> <mn>135</mn> <mi>M</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(M)\sim {\sqrt {\frac {\pi M}{2}}}-{\frac {1}{3}}+{\frac {1}{12}}{\sqrt {\frac {\pi }{2M}}}-{\frac {4}{135M}}+\cdots .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e7fdb17c8c1d5c328a63fee87762e0494378e46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:49.931ex; height:6.509ex;" alt="{\displaystyle Q(M)\sim {\sqrt {\frac {\pi M}{2}}}-{\frac {1}{3}}+{\frac {1}{12}}{\sqrt {\frac {\pi }{2M}}}-{\frac {4}{135M}}+\cdots .}" /></dd></dl> With M = 365 days in a year, the average number of people required to find a pair with the same birthday is n = 1 + Q(M) ≈ 24.61659, somewhat more than 23, the number required for a 50% chance. In the best case, two people will suffice; at worst, the maximum possible number of M + 1 = 366 people is needed; but on average, only 25 people are required An analysis using indicator random variables can provide a simpler but approximate analysis of this problem.<a href="#cite_note-26">[25]</a> For each pair (i, j) for k people in a room, we define the indicator random variable Xij, for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq i\leq j\leq k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>j</mi> <mo>≤</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq i\leq j\leq k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d842416cc2839dea2baa82259fe2c80086e5c650" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.43ex; height:2.509ex;" alt="{\displaystyle 1\leq i\leq j\leq k}" />, by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{2}X_{ij}&=I\{{\text{person }}i{\text{ and person }}j{\text{ have the same birthday}}\}\\[10pt]&={\begin{cases}1,&{\text{if person }}i{\text{ and person }}j{\text{ have the same birthday;}}\\0,&{\text{otherwise.}}\end{cases}}\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left" rowspacing="1.3em 0.3em" columnspacing="0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>I</mi> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>person </mtext> </mrow> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> and person </mtext> </mrow> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> have the same birthday</mtext> </mrow> <mo fence="false" stretchy="false">}</mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if person </mtext> </mrow> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> and person </mtext> </mrow> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> have the same birthday;</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise.</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{2}X_{ij}&=I\{{\text{person }}i{\text{ and person }}j{\text{ have the same birthday}}\}\\[10pt]&={\begin{cases}1,&{\text{if person }}i{\text{ and person }}j{\text{ have the same birthday;}}\\0,&{\text{otherwise.}}\end{cases}}\end{alignedat}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a46ca36a316dcaa896eb27ea485e183cfe67819e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:61.555ex; height:11.509ex;" alt="{\displaystyle {\begin{alignedat}{2}X_{ij}&=I\{{\text{person }}i{\text{ and person }}j{\text{ have the same birthday}}\}\\[10pt]&={\begin{cases}1,&{\text{if person }}i{\text{ and person }}j{\text{ have the same birthday;}}\\0,&{\text{otherwise.}}\end{cases}}\end{alignedat}}}" /> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{2}E[X_{ij}]&=\Pr\{{\text{person }}i{\text{ and person }}j{\text{ have the same birthday}}\}={\frac {1}{n}}.\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>person </mtext> </mrow> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> and person </mtext> </mrow> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> have the same birthday</mtext> </mrow> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{2}E[X_{ij}]&=\Pr\{{\text{person }}i{\text{ and person }}j{\text{ have the same birthday}}\}={\frac {1}{n}}.\end{alignedat}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/500efcea4d3c02589368f6bc6028b676f9a624d8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.79ex; margin-bottom: -0.215ex; width:66.207ex; height:5.176ex;" alt="{\displaystyle {\begin{alignedat}{2}E[X_{ij}]&=\Pr\{{\text{person }}i{\text{ and person }}j{\text{ have the same birthday}}\}={\frac {1}{n}}.\end{alignedat}}}" /> Let X be a random variable counting the pairs of individuals with the same birthday. <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\sum _{i=1}^{k}\sum _{j=i+1}^{k}X_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</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>k</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\sum _{i=1}^{k}\sum _{j=i+1}^{k}X_{ij}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0d30c9f0d77a2f45a7bc0eabf237f01cb3f8643" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:17.233ex; height:7.676ex;" alt="{\displaystyle X=\sum _{i=1}^{k}\sum _{j=i+1}^{k}X_{ij}}" /> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{3}E[X]&=\sum _{i=1}^{k}\sum _{j=i+1}^{k}E[X_{ij}]\\[8pt]&={\binom {k}{2}}{\frac {1}{n}}\\[8pt]&={\frac {k(k-1)}{2n}}\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left" rowspacing="1.1em 1.1em 0.3em" columnspacing="0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mi>E</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></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>k</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>k</mi> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{3}E[X]&=\sum _{i=1}^{k}\sum _{j=i+1}^{k}E[X_{ij}]\\[8pt]&={\binom {k}{2}}{\frac {1}{n}}\\[8pt]&={\frac {k(k-1)}{2n}}\end{alignedat}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55dd1b77df250394f00925d42e5bd67896842481" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.913ex; margin-bottom: -0.258ex; width:24.123ex; height:23.509ex;" alt="{\displaystyle {\begin{alignedat}{3}E[X]&=\sum _{i=1}^{k}\sum _{j=i+1}^{k}E[X_{ij}]\\[8pt]&={\binom {k}{2}}{\frac {1}{n}}\\[8pt]&={\frac {k(k-1)}{2n}}\end{alignedat}}}" /> For n = 365, if k = 28, the expected number of pairs of individuals with the same birthday is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠28 × 27/2 × 365⁠ ≈ 1.0356. Therefore, we can expect at least one matching pair with at least 28 people. In the <a href="/wiki/2014_FIFA_World_Cup" title="2014 FIFA World Cup">2014 FIFA World Cup</a>, each of the 32 squads had 23 players. An analysis of the official squad lists suggested that 16 squads had pairs of players sharing birthdays, and of these 5 squads had two pairs: Argentina, France, Iran, South Korea and Switzerland each had two pairs, and Australia, Bosnia and Herzegovina, Brazil, Cameroon, Colombia, Honduras, Netherlands, Nigeria, Russia, Spain and USA each with one pair.<a href="#cite_note-27">[26]</a> Voracek, Tran and <a href="/wiki/Anton_Formann" title="Anton Formann">Formann</a> showed that the majority of people markedly overestimate the number of people that is necessary to achieve a given probability of people having the same birthday, and markedly underestimate the probability of people having the same birthday when a specific sample size is given.<a href="#cite_note-28">[27]</a> Further results showed that psychology students and women did better on the task than casino visitors/personnel or men, but were less confident about their estimates. <div class="mw-heading mw-heading3"><h3 id="Reverse_problem">Reverse problem</h3>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=27" title="Edit section: Reverse problem">edit</a>]</div> The reverse problem is to find, for a fixed probability p, the greatest n for which the probability p(n) is smaller than the given p, or the smallest n for which the probability p(n) is greater than the given p.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] Taking the above formula for d = 365, one has <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n(p;365)\approx {\sqrt {730\ln \left({\frac {1}{1-p}}\right)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>;</mo> <mn>365</mn> <mo stretchy="false">)</mo> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>730</mn> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n(p;365)\approx {\sqrt {730\ln \left({\frac {1}{1-p}}\right)}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8198dd431a622bd448ef0a9f44fc8d23e43cc95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:30.207ex; height:7.509ex;" alt="{\displaystyle n(p;365)\approx {\sqrt {730\ln \left({\frac {1}{1-p}}\right)}}.}" /></dd></dl> The following table gives some sample calculations. <dl><dd><table class="wikitable"> <tbody><tr> <th>p</th> <th>n </th> <th>n↓</th> <th>p(n↓)</th> <th>n↑</th> <th>p(n↑) </th></tr> <tr> <td>0.01 </td> <td>0.14178√365 = 2.70864 </td> <td align="right">2</td> <td>0.00274</td> <td align="right">3 </td> <td>0.00820 </td></tr> <tr> <td>0.05</td> <td>0.32029√365 = 6.11916 </td> <td align="right">6</td> <td>0.04046</td> <td align="right">7</td> <td>0.05624 </td></tr> <tr> <td>0.1 </td> <td>0.45904√365 = 8.77002 </td> <td align="right">8</td> <td>0.07434</td> <td align="right">9 </td> <td>0.09462 </td></tr> <tr> <td>0.2 </td> <td>0.66805√365 = 12.76302 </td> <td align="right">12</td> <td>0.16702</td> <td align="right">13 </td> <td>0.19441 </td></tr> <tr> <td>0.3</td> <td>0.84460√365 = 16.13607 </td> <td align="right">16</td> <td>0.28360</td> <td align="right">17</td> <td>0.31501 </td></tr> <tr> <td>0.5</td> <td>1.17741√365 = 22.49439 </td> <td align="right">22</td> <td>0.47570</td> <td align="right">23</td> <td>0.50730 </td></tr> <tr> <td>0.7</td> <td>1.55176√365 = 29.64625 </td> <td align="right">29</td> <td>0.68097</td> <td align="right">30</td> <td>0.70632 </td></tr> <tr> <td>0.8</td> <td>1.79412√365 = 34.27666 </td> <td align="right">34</td> <td>0.79532</td> <td align="right">35</td> <td>0.81438 </td></tr> <tr> <td>0.9</td> <td>2.14597√365 = 40.99862 </td> <td align="right">40</td> <td>0.89123</td> <td align="right">41</td> <td>0.90315 </td></tr> <tr> <td>0.95</td> <td>2.44775√365 = 46.76414 </td> <td align="right">46</td> <td>0.94825</td> <td align="right">47</td> <td>0.95477 </td></tr> <tr> <td>0.99 </td> <td>3.03485√365 = 57.98081 </td> <td align="right">57 </td> <td>0.99012 </td> <td align="right">58</td> <td>0.99166 </td></tr></tbody></table></dd></dl> Some values falling outside the bounds have been colored to show that the approximation is not always exact. <div class="mw-heading mw-heading2"><h2 id="Partition_problem">Partition problem</h2>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=28" title="Edit section: Partition problem">edit</a>]</div> A related problem is the <a href="/wiki/Partition_problem" title="Partition problem">partition problem</a>, a variant of the <a href="/wiki/Knapsack_problem" title="Knapsack problem">knapsack problem</a> from <a href="/wiki/Operations_research" title="Operations research">operations research</a>. Some weights are put on a <a href="/wiki/Weighing_scale" title="Weighing scale">balance scale</a>; each weight is an integer number of grams randomly chosen between one gram and one million grams (one <a href="/wiki/Tonne" title="Tonne">tonne</a>). The question is whether one can usually (that is, with probability close to 1) transfer the weights between the left and right arms to balance the scale. (In case the sum of all the weights is an odd number of grams, a discrepancy of one gram is allowed.) If there are only two or three weights, the answer is very clearly no; although there are some combinations which work, the majority of randomly selected combinations of three weights do not. If there are very many weights, the answer is clearly yes. The question is, how many are just sufficient? That is, what is the number of weights such that it is equally likely for it to be possible to balance them as it is to be impossible? Often, people's intuition is that the answer is above 100000. Most people's intuition is that it is in the thousands or tens of thousands, while others feel it should at least be in the hundreds. The correct answer is 23.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] The reason is that the correct comparison is to the number of partitions of the weights into left and right. There are 2N − 1 different partitions for N weights, and the left sum minus the right sum can be thought of as a new random quantity for each partition. The distribution of the sum of weights is approximately <a href="/wiki/Normal_distribution" title="Normal distribution">Gaussian</a>, with a peak at 500000N and width 1000000√N, so that when 2N − 1 is approximately equal to 1000000√N the transition occurs. 223 − 1 is about 4 million, while the width of the distribution is only 5 million.<a href="#cite_note-29">[28]</a> <div class="mw-heading mw-heading2"><h2 id="In_fiction">In fiction</h2>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=29" title="Edit section: In fiction">edit</a>]</div> <a href="/wiki/Arthur_C._Clarke" title="Arthur C. Clarke">Arthur C. Clarke</a>'s 1961 novel <a href="/wiki/A_Fall_of_Moondust" title="A Fall of Moondust">A Fall of Moondust</a> contains a section where the main characters, trapped underground for an indefinite amount of time, are celebrating a birthday and find themselves discussing the validity of the birthday problem. As stated by a physicist passenger: "If you have a group of more than twenty-four people, the odds are better than even that two of them have the same birthday." Eventually, out of 22 present, it is revealed that two characters share the same birthday, May 23. <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=30" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-10"><a href="#cite_ref-10">^</a> In his autobiography, Halmos criticized the form in which the birthday paradox is often presented, in terms of numerical computation. He believed that it should be used as an example in the use of more abstract mathematical concepts. He wrote: <blockquote>The reasoning is based on important tools that all students of mathematics should have ready access to. The birthday problem used to be a splendid illustration of the advantages of pure thought over mechanical manipulation; the inequalities can be obtained in a minute or two, whereas the multiplications would take much longer, and be much more subject to error, whether the instrument is a pencil or an old-fashioned desk computer. What <a href="/wiki/Calculator" title="Calculator">calculators</a> do not yield is understanding, or mathematical facility, or a solid basis for more advanced, generalized theories.</blockquote> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=31" title="Edit section: References">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626" /><div class="reflist reflist-columns references-column-width" style="column-width: 45em;"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <a href="/wiki/David_Singmaster" title="David Singmaster">David Singmaster</a>, Sources in Recreational Mathematics: An Annotated Bibliography, Eighth Preliminary Edition, 2004, <a rel="nofollow" class="external text" href="https://www.puzzlemuseum.com/singma/singma6/SOURCES/singma-sources-edn8-2004-03-19.htm#_Toc69534221">section 8.B</a> </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <a href="/wiki/H.S.M._Coxeter" class="mw-redirect" title="H.S.M. Coxeter">H.S.M. Coxeter</a>, "Mathematical Recreations and Essays, 11th edition", 1940, p 45, as reported in <a href="/wiki/I._J._Good" title="I. J. Good">I. J. Good</a>, Probability and the weighing of evidence, 1950, <a rel="nofollow" class="external text" href="https://archive.org/details/probabilityweigh0000good/page/38/mode/2up?q=same%20birthday">p. 38</a> </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> Richard Von Mises, "Über Aufteilungs- und Besetzungswahrscheinlichkeiten", Revue de la faculté des sciences de l'Université d'Istanbul 4:145-163, 1939, reprinted in <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFFrankGoldsteinKacPrager1964" class="citation book cs1">Frank, P.; Goldstein, S.; Kac, M.; Prager, W.; Szegö, G.; Birkhoff, G., eds. (1964). Selected Papers of Richard von Mises. Vol. 2. Providence, Rhode Island: Amer. Math. Soc. pp. 313–334.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Selected+Papers+of+Richard+von+Mises&rft.place=Providence%2C+Rhode+Island&rft.pages=%3Cspan+class%3D%22nowrap%22%3E313-%3C%2Fspan%3E334&rft.pub=Amer.+Math.+Soc.&rft.date=1964&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABirthday+problem" class="Z3988"> </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> see <a href="/wiki/Birthday#Distribution_through_the_year" title="Birthday">Birthday#Distribution through the year</a> </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> (<a href="#CITEREFBloom1973">Bloom 1973</a>) </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSteele2004" class="citation book cs1">Steele, J. 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Random Structures and Algorithms. 19 (3–4): 247–288. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Frsa.10004">10.1002/rsa.10004</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:6819493">6819493</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Random+Structures+and+Algorithms&rft.atitle=Phase+Transition+and+Finite+Size+Scaling+in+the+Integer+Partition+Problem&rft.volume=19&rft.issue=%3Cspan+class%3D%22nowrap%22%3E3%E2%80%93%3C%2Fspan%3E4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E247-%3C%2Fspan%3E288&rft.date=2001&rft_id=info%3Adoi%2F10.1002%2Frsa.10004&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A6819493%23id-name%3DS2CID&rft.aulast=Borgs&rft.aufirst=C.&rft.au=Chayes%2C+J.&rft.au=Pittel%2C+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABirthday+problem" class="Z3988"> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=32" title="Edit section: Bibliography">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation journal cs1">Abramson, M.; Moser, W. O. J. (1970). "More Birthday Surprises". <a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a>. 77 (8): 856–858. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2317022">10.2307/2317022</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2317022">2317022</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=More+Birthday+Surprises&rft.volume=77&rft.issue=8&rft.pages=%3Cspan+class%3D%22nowrap%22%3E856-%3C%2Fspan%3E858&rft.date=1970&rft_id=info%3Adoi%2F10.2307%2F2317022&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2317022%23id-name%3DJSTOR&rft.aulast=Abramson&rft.aufirst=M.&rft.au=Moser%2C+W.+O.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABirthday+problem" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBloom1973" class="citation journal cs1">Bloom, D. (1973). "A Birthday Problem". <a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a>. 80 (10): 1141–1142. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2318556">10.2307/2318556</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2318556">2318556</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=A+Birthday+Problem&rft.volume=80&rft.issue=10&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1141-%3C%2Fspan%3E1142&rft.date=1973&rft_id=info%3Adoi%2F10.2307%2F2318556&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2318556%23id-name%3DJSTOR&rft.aulast=Bloom&rft.aufirst=D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABirthday+problem" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation book cs1">Kemeny, John G.; Snell, J. Laurie; Thompson, Gerald (1957). Introduction to Finite Mathematics (First ed.).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Finite+Mathematics&rft.edition=First&rft.date=1957&rft.aulast=Kemeny&rft.aufirst=John+G.&rft.au=Snell%2C+J.+Laurie&rft.au=Thompson%2C+Gerald&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABirthday+problem" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation journal cs1">McKinney, E. H. (1966). "Generalized Birthday Problem". <a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a>. 73 (5): 385–387. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2315408">10.2307/2315408</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2315408">2315408</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=Generalized+Birthday+Problem&rft.volume=73&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E385-%3C%2Fspan%3E387&rft.date=1966&rft_id=info%3Adoi%2F10.2307%2F2315408&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2315408%23id-name%3DJSTOR&rft.aulast=McKinney&rft.aufirst=E.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABirthday+problem" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMosteller1962" class="citation journal cs1">Mosteller, F. (1962). "Understanding the Birthday Problem". <a href="/wiki/The_Mathematics_Teacher" class="mw-redirect" title="The Mathematics Teacher">The Mathematics Teacher</a>. 55 (5): 322–325. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5951%2FMT.55.5.0322">10.5951/MT.55.5.0322</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/27956609">27956609</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematics+Teacher&rft.atitle=Understanding+the+Birthday+Problem&rft.volume=55&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E322-%3C%2Fspan%3E325&rft.date=1962&rft_id=info%3Adoi%2F10.5951%2FMT.55.5.0322&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F27956609%23id-name%3DJSTOR&rft.aulast=Mosteller&rft.aufirst=F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABirthday+problem" class="Z3988"> Reprinted in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMosteller2006" class="citation book cs1">Mosteller, Frederick (2006). "Understanding the Birthday Problem". Selected Papers of Frederick Mosteller. Springer Series in Statistics. pp. 349–353. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-0-387-44956-2_21">10.1007/978-0-387-44956-2_21</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-20271-6" title="Special:BookSources/978-0-387-20271-6"><bdi>978-0-387-20271-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Understanding+the+Birthday+Problem&rft.btitle=Selected+Papers+of+Frederick+Mosteller&rft.series=Springer+Series+in+Statistics&rft.pages=%3Cspan+class%3D%22nowrap%22%3E349-%3C%2Fspan%3E353&rft.date=2006&rft_id=info%3Adoi%2F10.1007%2F978-0-387-44956-2_21&rft.isbn=978-0-387-20271-6&rft.aulast=Mosteller&rft.aufirst=Frederick&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABirthday+problem" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation book cs1"><a href="/wiki/Leila_Schneps" title="Leila Schneps">Schneps, Leila</a>; <a href="/wiki/Coralie_Colmez" title="Coralie Colmez">Colmez, Coralie</a> (2013). "Math error number 5. The case of Diana Sylvester: cold hit analysis". <a href="/wiki/Math_on_Trial" title="Math on Trial">Math on Trial. How Numbers Get Used and Abused in the Courtroom</a>. Basic Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-465-03292-1" title="Special:BookSources/978-0-465-03292-1"><bdi>978-0-465-03292-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Math+error+number+5.+The+case+of+Diana+Sylvester%3A+cold+hit+analysis&rft.btitle=Math+on+Trial.+How+Numbers+Get+Used+and+Abused+in+the+Courtroom&rft.pub=Basic+Books&rft.date=2013&rft.isbn=978-0-465-03292-1&rft.aulast=Schneps&rft.aufirst=Leila&rft.au=Colmez%2C+Coralie&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABirthday+problem" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation book cs1">Sy M. Blinder (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=M7TCNAEACAAJ">Guide to Essential Math: A Review for Physics, Chemistry and Engineering Students</a>. Elsevier. pp. 5–6. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-407163-6" title="Special:BookSources/978-0-12-407163-6"><bdi>978-0-12-407163-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Guide+to+Essential+Math%3A+A+Review+for+Physics%2C+Chemistry+and+Engineering+Students&rft.pages=%3Cspan+class%3D%22nowrap%22%3E5-%3C%2Fspan%3E6&rft.pub=Elsevier&rft.date=2013&rft.isbn=978-0-12-407163-6&rft.au=Sy+M.+Blinder&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DM7TCNAEACAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABirthday+problem" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Birthday_problem&action=edit&section=33" title="Edit section: External links">edit</a>]</div> <ul><li><a rel="nofollow" class="external text" href="http://www.efgh.com/math/birthday.htm">The Birthday Paradox accounting for leap year birthdays</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite 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/BirthdayProblem.html">"Birthday Problem"</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</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=Birthday+Problem&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FBirthdayProblem.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABirthday+problem" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="http://www.damninteresting.com/?p=402">A humorous article explaining the paradox</a></li> <li><a rel="nofollow" class="external text" href="http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_BirthdayExperiment">SOCR EduMaterials activities birthday experiment</a></li> <li><a rel="nofollow" class="external text" href="http://betterexplained.com/articles/understanding-the-birthday-paradox/">Understanding the Birthday Problem (Better Explained)</a></li> <li><a rel="nofollow" class="external text" href="http://www.matifutbol.com/en/eurobirthdays.html">Eurobirthdays 2012. A birthday problem.</a> A practical football example of the birthday paradox.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1">Grime, James. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170225140726/http://www.numberphile.com/videos/23birthday.html">"23: Birthday Probability"</a>. Numberphile. <a href="/wiki/Brady_Haran" title="Brady Haran">Brady Haran</a>. Archived from <a rel="nofollow" class="external text" href="http://www.numberphile.com/videos/23birthday.html">the original</a> on 2017-02-25. Retrieved 2013-04-02.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Numberphile&rft.atitle=23%3A+Birthday+Probability&rft.aulast=Grime&rft.aufirst=James&rft_id=http%3A%2F%2Fwww.numberphile.com%2Fvideos%2F23birthday.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABirthday+problem" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="https://www.wolframalpha.com/input/?i=birthday+paradox%2C+4+people%2C+100+possible+birthdays">Computing the probabilities of the Birthday Problem at WolframAlpha</a></li></ul> <style data-mw-deduplicate="TemplateStyles:r1130092004">.mw-parser-output .portal-bar{font-size:88%;font-weight:bold;display:flex;justify-content:center;align-items:baseline}.mw-parser-output .portal-bar-bordered{padding:0 2em;background-color:#fdfdfd;border:1px solid #a2a9b1;clear:both;margin:1em auto 0}.mw-parser-output .portal-bar-related{font-size:100%;justify-content:flex-start}.mw-parser-output .portal-bar-unbordered{padding:0 1.7em;margin-left:0}.mw-parser-output .portal-bar-header{margin:0 1em 0 0.5em;flex:0 0 auto;min-height:24px}.mw-parser-output .portal-bar-content{display:flex;flex-flow:row wrap;flex:0 1 auto;padding:0.15em 0;column-gap:1em;align-items:baseline;margin:0;list-style:none}.mw-parser-output .portal-bar-content-related{margin:0;list-style:none}.mw-parser-output .portal-bar-item{display:inline-block;margin:0.15em 0.2em;min-height:24px;line-height:24px}@media screen and (max-width:768px){.mw-parser-output .portal-bar{font-size:88%;font-weight:bold;display:flex;flex-flow:column wrap;align-items:baseline}.mw-parser-output .portal-bar-header{text-align:center;flex:0;padding-left:0.5em;margin:0 auto}.mw-parser-output .portal-bar-related{font-size:100%;align-items:flex-start}.mw-parser-output .portal-bar-content{display:flex;flex-flow:row wrap;align-items:center;flex:0;column-gap:1em;border-top:1px solid #a2a9b1;margin:0 auto;list-style:none}.mw-parser-output .portal-bar-content-related{border-top:none;margin:0;list-style:none}}.mw-parser-output .navbox+link+.portal-bar,.mw-parser-output .navbox+style+.portal-bar,.mw-parser-output .navbox+link+.portal-bar-bordered,.mw-parser-output .navbox+style+.portal-bar-bordered,.mw-parser-output .sister-bar+link+.portal-bar,.mw-parser-output .sister-bar+style+.portal-bar,.mw-parser-output .portal-bar+.navbox-styles+.navbox,.mw-parser-output .portal-bar+.navbox-styles+.sister-bar{margin-top:-1px}</style><div class="portal-bar noprint metadata noviewer portal-bar-bordered" role="navigation" aria-label="Portals"><a href="/wiki/Wikipedia:Contents/Portals" title="Wikipedia:Contents/Portals">Portal</a>:<ul class="portal-bar-content"><li class="portal-bar-item"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/19px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="19" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/29px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/38px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a> <a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics</a></li></ul></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; 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Birthday problem - Wikipedia