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vector-toc-level-2"> <a class="vector-toc-link" href="#Rolling_dice"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Rolling dice</span> </div> </a> <ul id="toc-Rolling_dice-sublist" class="vector-toc-list"> <li id="toc-Solution_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Solution_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Solution</span> </div> </a> <ul id="toc-Solution_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a 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class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Assaig_de_Bernoulli" title="Assaig de Bernoulli – Catalan" lang="ca" hreflang="ca" data-title="Assaig de Bernoulli" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Bernoulliho_pokus" title="Bernoulliho pokus – Czech" lang="cs" hreflang="cs" data-title="Bernoulliho pokus" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ensayo_de_Bernoulli" title="Ensayo de Bernoulli – Spanish" lang="es" hreflang="es" data-title="Ensayo de Bernoulli" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Provo_de_Bernoulli" title="Provo de Bernoulli – Esperanto" lang="eo" hreflang="eo" data-title="Provo de Bernoulli" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Bernoulli_saiakuntza" title="Bernoulli saiakuntza – Basque" lang="eu" hreflang="eu" data-title="Bernoulli saiakuntza" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A2%D8%B2%D9%85%D8%A7%DB%8C%D8%B4_%D8%A8%D8%B1%D9%86%D9%88%D9%84%DB%8C" title="آزمایش برنولی – Persian" lang="fa" hreflang="fa" data-title="آزمایش برنولی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/%C3%89preuve_de_Bernoulli" title="Épreuve de Bernoulli – French" lang="fr" hreflang="fr" data-title="Épreuve de Bernoulli" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B2%A0%EB%A5%B4%EB%88%84%EC%9D%B4_%EC%8B%9C%ED%96%89" title="베르누이 시행 – Korean" lang="ko" hreflang="ko" data-title="베르누이 시행" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Bernoulli-teszt" title="Bernoulli-teszt – Hungarian" lang="hu" hreflang="hu" data-title="Bernoulli-teszt" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%91%D0%B5%D1%80%D0%BD%D1%83%D0%BB%D0%B8%D0%B5%D0%B2_%D0%BE%D0%BF%D0%B8%D1%82" title="Бернулиев опит – Macedonian" lang="mk" hreflang="mk" data-title="Бернулиев опит" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%91%D0%B5%D1%80%D0%BD%D1%83%D0%BB%D0%BB%D0%B8%D0%B9%D0%BD_%D1%82%D1%83%D1%80%D1%88%D0%B8%D0%BB%D1%82" title="Бернуллийн туршилт – Mongolian" lang="mn" hreflang="mn" data-title="Бернуллийн туршилт" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Bernoulli-experiment" title="Bernoulli-experiment – Dutch" lang="nl" hreflang="nl" data-title="Bernoulli-experiment" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%99%E3%83%AB%E3%83%8C%E3%83%BC%E3%82%A4%E8%A9%A6%E8%A1%8C" title="ベルヌーイ試行 – Japanese" lang="ja" hreflang="ja" data-title="ベルヌーイ試行" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Bernoulli-fors%C3%B8k" title="Bernoulli-forsøk – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Bernoulli-forsøk" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Preuva_%C3%ABd_Bernoulli" title="Preuva ëd Bernoulli – Piedmontese" lang="pms" hreflang="pms" data-title="Preuva ëd Bernoulli" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Pr%C3%B3ba_Bernoulliego" title="Próba Bernoulliego – Polish" lang="pl" hreflang="pl" data-title="Próba Bernoulliego" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Ensaio_de_Bernoulli" title="Ensaio de Bernoulli – Portuguese" lang="pt" hreflang="pt" data-title="Ensaio de Bernoulli" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Bernoullijev_poskus" title="Bernoullijev poskus – Slovenian" lang="sl" hreflang="sl" data-title="Bernoullijev poskus" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Binomitodenn%C3%A4k%C3%B6isyys" title="Binomitodennäköisyys – Finnish" lang="fi" hreflang="fi" data-title="Binomitodennäköisyys" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A1%D1%85%D0%B5%D0%BC%D0%B0_%D0%91%D0%B5%D1%80%D0%BD%D1%83%D0%BB%D0%BB%D1%96" title="Схема Бернуллі – Ukrainian" lang="uk" hreflang="uk" data-title="Схема Бернуллі" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A9p_th%E1%BB%AD_Bernoulli" title="Phép thử Bernoulli – Vietnamese" lang="vi" hreflang="vi" data-title="Phép thử Bernoulli" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E4%BC%AF%E5%8A%AA%E5%88%A9%E8%A9%A6%E9%A9%97" title="伯努利試驗 – Cantonese" lang="yue" hreflang="yue" data-title="伯努利試驗" data-language-autonym="粵語" data-language-local-name="Cantonese" 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class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Standard_deviation_diagram_micro.svg/250px-Standard_deviation_diagram_micro.svg.png" decoding="async" width="250" height="125" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Standard_deviation_diagram_micro.svg/375px-Standard_deviation_diagram_micro.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Standard_deviation_diagram_micro.svg/500px-Standard_deviation_diagram_micro.svg.png 2x" data-file-width="400" data-file-height="200" /></a></span></td></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Probability" title="Probability">Probability</a> <ul><li><a href="/wiki/Probability_axioms" title="Probability axioms">Axioms</a></li></ul></li> <li><a href="/wiki/Determinism" title="Determinism">Determinism</a> <ul><li><a href="/wiki/Deterministic_system" title="Deterministic system">System</a></li></ul></li> <li><a href="/wiki/Indeterminism" title="Indeterminism">Indeterminism</a></li> <li><a href="/wiki/Randomness" title="Randomness">Randomness</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Probability_space" title="Probability space">Probability space</a></li> <li><a href="/wiki/Sample_space" title="Sample space">Sample space</a></li> <li><a href="/wiki/Event_(probability_theory)" title="Event (probability theory)">Event</a> <ul><li><a href="/wiki/Collectively_exhaustive_events" title="Collectively exhaustive events">Collectively exhaustive events</a></li> <li><a href="/wiki/Elementary_event" title="Elementary event">Elementary event</a></li> <li><a href="/wiki/Mutual_exclusivity" title="Mutual exclusivity">Mutual exclusivity</a></li> <li><a href="/wiki/Outcome_(probability)" title="Outcome (probability)">Outcome</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li></ul></li> <li><a href="/wiki/Experiment_(probability_theory)" title="Experiment (probability theory)">Experiment</a> <ul><li><a class="mw-selflink selflink">Bernoulli trial</a></li></ul></li> <li><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distribution</a> <ul><li><a href="/wiki/Bernoulli_distribution" title="Bernoulli distribution">Bernoulli distribution</a></li> <li><a href="/wiki/Binomial_distribution" title="Binomial distribution">Binomial distribution</a></li> <li><a href="/wiki/Exponential_distribution" title="Exponential distribution">Exponential distribution</a></li> <li><a href="/wiki/Normal_distribution" title="Normal distribution">Normal distribution</a></li> <li><a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto distribution</a></li> <li><a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson distribution</a></li></ul></li> <li><a href="/wiki/Probability_measure" title="Probability measure">Probability measure</a></li> <li><a href="/wiki/Random_variable" title="Random variable">Random variable</a> <ul><li><a href="/wiki/Bernoulli_process" title="Bernoulli process">Bernoulli process</a></li> <li><a href="/wiki/Continuous_or_discrete_variable" title="Continuous or discrete variable">Continuous or discrete</a></li> <li><a href="/wiki/Expected_value" title="Expected value">Expected value</a></li> <li><a href="/wiki/Variance" title="Variance">Variance</a></li> <li><a href="/wiki/Markov_chain" title="Markov chain">Markov chain</a></li> <li><a href="/wiki/Realization_(probability)" title="Realization (probability)">Observed value</a></li> <li><a href="/wiki/Random_walk" title="Random walk">Random walk</a></li> <li><a href="/wiki/Stochastic_process" title="Stochastic process">Stochastic process</a></li></ul></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Complementary_event" title="Complementary event">Complementary event</a></li> <li><a href="/wiki/Joint_probability_distribution" title="Joint probability distribution">Joint probability</a></li> <li><a href="/wiki/Marginal_distribution" title="Marginal distribution">Marginal probability</a></li> <li><a href="/wiki/Conditional_probability" title="Conditional probability">Conditional probability</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Independence_(probability_theory)" title="Independence (probability theory)">Independence</a></li> <li><a href="/wiki/Conditional_independence" title="Conditional independence">Conditional independence</a></li> <li><a href="/wiki/Law_of_total_probability" title="Law of total probability">Law of total probability</a></li> <li><a href="/wiki/Law_of_large_numbers" title="Law of large numbers">Law of large numbers</a></li> <li><a href="/wiki/Bayes%27_theorem" title="Bayes' theorem">Bayes' theorem</a></li> <li><a href="/wiki/Boole%27s_inequality" title="Boole's inequality">Boole's inequality</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li> <li><a href="/wiki/Tree_diagram_(probability_theory)" title="Tree diagram (probability theory)">Tree diagram</a></li></ul></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Probability_fundamentals" title="Template:Probability fundamentals"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Probability_fundamentals" title="Template talk:Probability fundamentals"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Probability_fundamentals" title="Special:EditPage/Template:Probability fundamentals"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Bernoulli_trial_progression.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Bernoulli_trial_progression.svg/400px-Bernoulli_trial_progression.svg.png" decoding="async" width="400" height="400" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Bernoulli_trial_progression.svg/600px-Bernoulli_trial_progression.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Bernoulli_trial_progression.svg/800px-Bernoulli_trial_progression.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>Graphs of probability <i>P</i> of not observing independent events each of probability <i>p</i> after <i>n</i> Bernoulli trials vs <i>np</i> for various <i>p</i>. Three examples are shown: <br /><b>Blue curve</b>: Throwing a 6-sided die 6 times gives a 33.5% chance that 6 (or any other given number) never turns up; it can be observed that as <i>n</i> increases, the probability of a 1/<i>n</i>-chance event never appearing after <i>n</i> tries rapidly converges to <i>0</i>. <br /><b>Grey curve</b>: To get 50-50 chance of throwing a <a href="/wiki/Yahtzee" title="Yahtzee">Yahtzee</a> (5 cubic dice all showing the same number) requires 0.69 × 1296 ~ 898 throws. <br /><b>Green curve</b>: Drawing a card from a deck of playing cards without jokers 100 (1.92 × 52) times with replacement gives 85.7% chance of drawing the ace of spades at least once.</figcaption></figure> <p>In the theory of <a href="/wiki/Probability" title="Probability">probability</a> and <a href="/wiki/Statistics" title="Statistics">statistics</a>, a <b>Bernoulli trial</b> (or <b>binomial trial</b>) is a random <a href="/wiki/Experiment_(probability_theory)" title="Experiment (probability theory)">experiment</a> with exactly two possible <a href="/wiki/Outcome_(probability)" title="Outcome (probability)">outcomes</a>, "success" and "failure", in which the probability of success is the same every time the experiment is conducted.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> It is named after <a href="/wiki/Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a>, a 17th-century Swiss mathematician, who analyzed them in his <i><span title="Latin-language text"><i lang="la"><a href="/wiki/Ars_Conjectandi" title="Ars Conjectandi">Ars Conjectandi</a></i></span></i> (1713).<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>The mathematical formalization and advanced formulation of the Bernoulli trial is known as the <a href="/wiki/Bernoulli_process" title="Bernoulli process">Bernoulli process</a>. </p><p>Since a Bernoulli trial has only two possible outcomes, it can be framed as a "yes or no" question. For example: </p> <ul><li>Is the top card of a shuffled deck an ace?</li> <li>Was the newborn child a girl? (See <a href="/wiki/Human_sex_ratio" title="Human sex ratio">human sex ratio</a>.)</li></ul> <p>Success and failure are in this context labels for the two outcomes, and should not be construed literally or as value judgments. More generally, given any <a href="/wiki/Probability_space" title="Probability space">probability space</a>, for any <a href="/wiki/Event_(probability_theory)" title="Event (probability theory)">event</a> (set of outcomes), one can define a Bernoulli trial according to whether the event occurred or not (event or <a href="/wiki/Complementary_event" title="Complementary event">complementary event</a>). Examples of Bernoulli trials include: </p> <ul><li><a href="/wiki/Flipping_a_coin" class="mw-redirect" title="Flipping a coin">Flipping a coin</a>. In this context, obverse ("heads") conventionally denotes success and reverse ("tails") denotes failure. A <a href="/wiki/Fair_coin" title="Fair coin">fair coin</a> has the probability of success 0.5 by definition. In this case, there are exactly two possible outcomes.</li> <li>Rolling a <a href="/wiki/Dice" title="Dice">die</a>, where a six is "success" and everything else a "failure". In this case, there are six possible outcomes, and the event is a six; the complementary event "not a six" corresponds to the other five possible outcomes.</li> <li>In conducting a political <a href="/wiki/Opinion_poll" title="Opinion poll">opinion poll</a>, choosing a voter at random to ascertain whether that voter will vote "yes" in an upcoming referendum.</li></ul> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bernoulli_trial&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Independent repeated trials of an experiment with exactly two possible outcomes are called Bernoulli trials. Call one of the outcomes "success" and the other outcome "failure". Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> be the probability of success in a Bernoulli trial, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> be the probability of failure. Then the probability of success and the probability of failure sum to one, since these are complementary events: "success" and "failure" are <a href="/wiki/Mutually_exclusive" class="mw-redirect" title="Mutually exclusive">mutually exclusive</a> and <a href="/wiki/Collectively_exhaustive_events" title="Collectively exhaustive events">exhaustive</a>. Thus, one has the following relations: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=1-q,\quad \quad q=1-p,\quad \quad p+q=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>q</mi> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mi>q</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mi>p</mi> <mo>+</mo> <mi>q</mi> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=1-q,\quad \quad q=1-p,\quad \quad p+q=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/019911e57ce5071f70cfb113cd336fb25d223534" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:40.115ex; height:2.509ex;" alt="{\displaystyle p=1-q,\quad \quad q=1-p,\quad \quad p+q=1.}"></span></dd></dl> <p>Alternatively, these can be stated in terms of <a href="/wiki/Odds_(statistics)" class="mw-redirect" title="Odds (statistics)">odds</a>: given probability <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span></i> of success and <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span></i> of failure, the <i>odds for</i> are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>:</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p:q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db71c7b8bd28932bafd661e4d4f1bad9107b97cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.266ex; height:2.009ex;" alt="{\displaystyle p:q}"></span> and the <i>odds against</i> are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q:p.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>:</mo> <mi>p</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q:p.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e50d8e41857704cc2e92adb5a1e9df2ea4514f5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.823ex; height:2.009ex;" alt="{\displaystyle q:p.}"></span> These can also be expressed as numbers, by dividing, yielding the odds for, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o_{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>o</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o_{f}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab8863a66ff56f82b8ffee929dee2d6576a9e1a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.264ex; height:2.343ex;" alt="{\displaystyle o_{f}}"></span>, and the odds against, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o_{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>o</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o_{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a35966c6a92f117045187744cdc1412100e74f39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.229ex; height:2.009ex;" alt="{\displaystyle o_{a}}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}o_{f}&=p/q=p/(1-p)=(1-q)/q\\o_{a}&=q/p=(1-p)/p=q/(1-q).\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> <msub> <mi>o</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> <mo>=</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>o</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo>=</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}o_{f}&=p/q=p/(1-p)=(1-q)/q\\o_{a}&=q/p=(1-p)/p=q/(1-q).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67b8e4ec147a11d017bd1f8ecbe57b36c687d4f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.786ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}o_{f}&=p/q=p/(1-p)=(1-q)/q\\o_{a}&=q/p=(1-p)/p=q/(1-q).\end{aligned}}}"></span></dd></dl> <p>These are <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverses</a>, so they multiply to 1, with the following relations: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o_{f}=1/o_{a},\quad o_{a}=1/o_{f},\quad o_{f}\cdot o_{a}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>o</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>o</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <msub> <mi>o</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>o</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <msub> <mi>o</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>o</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o_{f}=1/o_{a},\quad o_{a}=1/o_{f},\quad o_{f}\cdot o_{a}=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4c118526deee783decd5ff67785789be0a93aab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.627ex; height:3.009ex;" alt="{\displaystyle o_{f}=1/o_{a},\quad o_{a}=1/o_{f},\quad o_{f}\cdot o_{a}=1.}"></span></dd></dl> <p>In the case that a Bernoulli trial is representing an event from finitely many <a href="/wiki/Equally_likely_outcomes" class="mw-redirect" title="Equally likely outcomes">equally likely outcomes</a>, where <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span></i> of the outcomes are success and <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span></i> of the outcomes are failure, the odds for are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S:F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>:</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S:F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14f4f36ffc919cba64b563b41b20e18d63972ad6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.177ex; height:2.176ex;" alt="{\displaystyle S:F}"></span> and the odds against are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:S.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>:</mo> <mi>S</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F:S.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f23bd168e81a4d976474766bc35d174a2c0fd8b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.824ex; height:2.176ex;" alt="{\displaystyle F:S.}"></span> This yields the following formulas for probability and odds: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}p&=S/(S+F)\\q&=F/(S+F)\\o_{f}&=S/F\\o_{a}&=F/S.\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> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo>+</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>q</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo>+</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>o</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>F</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>o</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>S</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p&=S/(S+F)\\q&=F/(S+F)\\o_{f}&=S/F\\o_{a}&=F/S.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff125fc92ba5652e5a066b50aff4dbf73ffa0d41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:16.907ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}p&=S/(S+F)\\q&=F/(S+F)\\o_{f}&=S/F\\o_{a}&=F/S.\end{aligned}}}"></span></dd></dl> <p>Here the odds are computed by dividing the number of outcomes, not the probabilities, but the proportion is the same, since these ratios only differ by multiplying both terms by the same constant factor. </p><p><a href="/wiki/Random_variable" title="Random variable">Random variables</a> describing Bernoulli trials are often encoded using the convention that 1 = "success", 0 = "failure". </p><p>Closely related to a Bernoulli trial is a binomial experiment, which consists of a fixed number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> of <a href="/wiki/Statistically_independent" class="mw-redirect" title="Statistically independent">statistically independent</a> Bernoulli trials, each with a probability of success <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>, and counts the number of successes. A random variable corresponding to a binomial experiment is denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(n,p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B(n,p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58280f6b0f1a1b474a7047c07943f908e775aa71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.171ex; height:2.843ex;" alt="{\displaystyle B(n,p)}"></span>, and is said to have a <i><a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial distribution</a></i>. The probability of exactly <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> successes in the experiment <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(n,p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B(n,p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58280f6b0f1a1b474a7047c07943f908e775aa71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.171ex; height:2.843ex;" alt="{\displaystyle B(n,p)}"></span> is given by: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(k)={n \choose k}p^{k}q^{n-k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <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> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(k)={n \choose k}p^{k}q^{n-k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a83905408846e50451f9a68554afdca80ecac090" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.371ex; height:6.176ex;" alt="{\displaystyle P(k)={n \choose k}p^{k}q^{n-k}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8cc51538192fdf193790d4378c3a998a6b94262" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:4.816ex; height:6.176ex;" alt="{\displaystyle {n \choose k}}"></span> is a <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficient</a>. </p><p>Bernoulli trials may also lead to <a href="/wiki/Negative_binomial_distribution" title="Negative binomial distribution">negative binomial distributions</a> (which count the number of successes in a series of repeated Bernoulli trials until a specified number of failures are seen), as well as various other distributions. </p><p>When multiple Bernoulli trials are performed, each with its own probability of success, these are sometimes referred to as <a href="/wiki/Poisson_trial" class="mw-redirect" title="Poisson trial">Poisson trials</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bernoulli_trial&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Tossing_coins">Tossing coins</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bernoulli_trial&action=edit&section=3" title="Edit section: Tossing coins"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider the simple experiment where a fair coin is tossed four times. Find the probability that exactly two of the tosses result in heads. </p> <div class="mw-heading mw-heading4"><h4 id="Solution">Solution</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bernoulli_trial&action=edit&section=4" title="Edit section: Solution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Coin_flip_outcomes.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Coin_flip_outcomes.png/220px-Coin_flip_outcomes.png" decoding="async" width="220" height="168" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Coin_flip_outcomes.png/330px-Coin_flip_outcomes.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Coin_flip_outcomes.png/440px-Coin_flip_outcomes.png 2x" data-file-width="1340" data-file-height="1022" /></a><figcaption>A representation of the possible outcomes of flipping a fair coin four times in terms of the number of heads. As can be seen, the probability of getting exactly two heads in four flips is 6/16 = 3/8, which matches the calculations.</figcaption></figure> <p>For this experiment, let a heads be defined as a <i>success</i> and a tails as a <i>failure.</i> Because the coin is assumed to be fair, the probability of success is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p={\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p={\tfrac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65a18fde17cf506242214406888b4a5975708380" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-left: -0.089ex; width:6.015ex; height:3.509ex;" alt="{\displaystyle p={\tfrac {1}{2}}}"></span>. Thus, the probability of failure, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>, is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=1-p=1-{\tfrac {1}{2}}={\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo>=</mo> <mn>1</mn> <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"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=1-p=1-{\tfrac {1}{2}}={\tfrac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ee590e199b7d1bb1ff2e57c1e591020c3a55397" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:22.856ex; height:3.509ex;" alt="{\displaystyle q=1-p=1-{\tfrac {1}{2}}={\tfrac {1}{2}}}"></span>.</dd></dl> <p>Using the equation above, the probability of exactly two tosses out of four total tosses resulting in a heads is given by: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}P(2)&={4 \choose 2}p^{2}q^{4-2}\\&=6\times \left({\tfrac {1}{2}}\right)^{2}\times \left({\tfrac {1}{2}}\right)^{2}\\&={\dfrac {3}{8}}.\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> <mn>2</mn> <mo stretchy="false">)</mo> </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"> <mn>4</mn> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>6</mn> <mo>×</mo> <msup> <mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>×</mo> <msup> <mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </mstyle> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P(2)&={4 \choose 2}p^{2}q^{4-2}\\&=6\times \left({\tfrac {1}{2}}\right)^{2}\times \left({\tfrac {1}{2}}\right)^{2}\\&={\dfrac {3}{8}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e455aebf9a65e7d5cca1c3a1cf67115c7716b1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:25.095ex; height:15.843ex;" alt="{\displaystyle {\begin{aligned}P(2)&={4 \choose 2}p^{2}q^{4-2}\\&=6\times \left({\tfrac {1}{2}}\right)^{2}\times \left({\tfrac {1}{2}}\right)^{2}\\&={\dfrac {3}{8}}.\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Rolling_dice">Rolling dice</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bernoulli_trial&action=edit&section=5" title="Edit section: Rolling dice"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>What is probability that when three independent fair six-sided dice are rolled, exactly two yield sixes? </p> <div class="mw-heading mw-heading4"><h4 id="Solution_2">Solution</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bernoulli_trial&action=edit&section=6" title="Edit section: Solution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/w/index.php?title=File:Binomial_trial_dice.svg&lang=egl" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Binomial_trial_dice.svg/langegl-220px-Binomial_trial_dice.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Binomial_trial_dice.svg/langegl-330px-Binomial_trial_dice.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Binomial_trial_dice.svg/langegl-440px-Binomial_trial_dice.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>Probabilities of rolling <i>k</i> sixes from <i>n</i> independent fair dice, with crossed out dice denoting non-six rolls – 2 sixes out of 3 dice is circled</figcaption></figure> <p>On one die, the probability of rolling a six, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p={\tfrac {1}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p={\tfrac {1}{6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c9f332df707c117d833267fc38767c49615cd70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; margin-left: -0.089ex; width:6.015ex; height:3.676ex;" alt="{\displaystyle p={\tfrac {1}{6}}}"></span>. Thus, the probability of not rolling a six, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=1-p={\tfrac {5}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5</mn> <mn>6</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=1-p={\tfrac {5}{6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76662c92a919392a8658ffaac5d59a8ae99734c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:14.097ex; height:3.676ex;" alt="{\displaystyle q=1-p={\tfrac {5}{6}}}"></span>. </p><p>As above, the probability of exactly two sixes out of three, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}P(2)&={3 \choose 2}p^{2}q^{3-2}\\&=3\times \left({\tfrac {1}{6}}\right)^{2}\times \left({\tfrac {5}{6}}\right)^{1}\\&={\dfrac {5}{72}}\approx 0.069.\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> <mn>2</mn> <mo stretchy="false">)</mo> </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"> <mn>3</mn> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>3</mn> <mo>×</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>×</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5</mn> <mn>6</mn> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>5</mn> <mn>72</mn> </mfrac> </mstyle> </mrow> <mo>≈</mo> <mn>0.069.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P(2)&={3 \choose 2}p^{2}q^{3-2}\\&=3\times \left({\tfrac {1}{6}}\right)^{2}\times \left({\tfrac {5}{6}}\right)^{1}\\&={\dfrac {5}{72}}\approx 0.069.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b922b55aaae21015c962895b18a60436892e189" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.838ex; width:26.386ex; height:16.843ex;" alt="{\displaystyle {\begin{aligned}P(2)&={3 \choose 2}p^{2}q^{3-2}\\&=3\times \left({\tfrac {1}{6}}\right)^{2}\times \left({\tfrac {5}{6}}\right)^{1}\\&={\dfrac {5}{72}}\approx 0.069.\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bernoulli_trial&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Bernoulli_scheme" title="Bernoulli scheme">Bernoulli scheme</a></li> <li><a href="/wiki/Bernoulli_sampling" title="Bernoulli sampling">Bernoulli sampling</a></li> <li><a href="/wiki/Bernoulli_distribution" title="Bernoulli distribution">Bernoulli distribution</a></li> <li><a href="/wiki/Binomial_distribution" title="Binomial distribution">Binomial distribution</a></li> <li><a href="/wiki/Binomial_coefficient" title="Binomial coefficient">Binomial coefficient</a></li> <li><a href="/wiki/Binomial_proportion_confidence_interval" title="Binomial proportion confidence interval">Binomial proportion confidence interval</a></li> <li><a href="/wiki/Poisson_sampling" title="Poisson sampling">Poisson sampling</a></li> <li><a href="/wiki/Sampling_design" title="Sampling design">Sampling design</a></li> <li><a href="/wiki/Coin_flipping" title="Coin flipping">Coin flipping</a></li> <li><a href="/wiki/Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a></li> <li><a href="/wiki/Fisher%27s_exact_test" title="Fisher's exact test">Fisher's exact test</a></li> <li><a href="/wiki/Boschloo%27s_test" title="Boschloo's test">Boschloo's test</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bernoulli_trial&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFPapoulis1984" class="citation encyclopaedia cs1">Papoulis, A. (1984). "Bernoulli Trials". <i>Probability, Random Variables, and Stochastic Processes</i> (2nd ed.). New York: <a href="/wiki/McGraw-Hill" class="mw-redirect" title="McGraw-Hill">McGraw-Hill</a>. pp. 57–63.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Bernoulli+Trials&rft.btitle=Probability%2C+Random+Variables%2C+and+Stochastic+Processes&rft.place=New+York&rft.pages=57-63&rft.edition=2nd&rft.pub=McGraw-Hill&rft.date=1984&rft.aulast=Papoulis&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernoulli+trial" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">James Victor Uspensky: <i>Introduction to Mathematical Probability</i>, McGraw-Hill, New York 1937, page 45</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="/wiki/Rajeev_Motwani" title="Rajeev Motwani">Rajeev Motwani</a> and P. Raghavan. Randomized Algorithms. Cambridge University Press, New York (NY), 1995, p.67-68</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bernoulli_trial&action=edit&section=9" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Bernoulli_trial" class="extiw" title="commons:Category:Bernoulli trial">Bernoulli trial</a></span>.</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Bernoulli_trials">"Bernoulli trials"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Bernoulli+trials&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DBernoulli_trials&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernoulli+trial" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.math.uah.edu/stat/applets/BinomialTimelineExperiment.html">"Simulation of n Bernoulli trials"</a>. math.uah.edu<span class="reference-accessdate">. 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