SherbyNet - /page/wikibook/entry.mu

<!DOCTYPE html>

<title>Probability</title>

</head>

<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Probability</span></span>

</h1>

</header>

/* start https://en.wikipedia.org/ */

.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}}

/* end https://en.wikipedia.org/ */

</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Probability_theory" title="Probability theory">probability theory</a>, <a href="Game_theory" title="Game theory">game theory</a>, <a href="Graph_theory" title="Graph theory">graph theory</a>, or <a href="Statistics" title="Statistics">statistics</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="Probability_(disambiguation)" class="mw-disambig" title="Probability (disambiguation)">Probability (disambiguation)</a>.</div>

</p>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1246091330">

/* start https://en.wikipedia.org/ */

.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><table class="sidebar nomobile nowraplinks hlist"><tbody><tr><th class="sidebar-title"></th></tr><tr><td class="sidebar-image"></td></tr><tr><td class="sidebar-content">

<ul><li><a href="Outline_of_probability" title="Outline of probability">Outline</a></li>

<li><a href="Catalog_of_articles_in_probability_theory" title="Catalog of articles in probability theory">Catalog of articles</a></li>

<li><a href="List_of_mathematical_probabilists" title="List of mathematical probabilists">Probabilists</a></li>

<li><a href="Glossary_of_probability_and_statistics" title="Glossary of probability and statistics">Glossary</a></li>

<li><a href="Notation_in_probability_and_statistics" title="Notation in probability and statistics">Notation</a></li>

<li><a href="List_of_probability_journals" title="List of probability journals">Journals</a></li>

<li>Category</li>

<li><br><span class="nowrap"><span class="skin-invert-image noviewer" typeof="mw:File"></span> </span><a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></li></ul></td>

</tr><tr><td class="sidebar-navbar"><style data-mw-deduplicate="TemplateStyles:r1239400231">

/* start https://en.wikipedia.org/ */

.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}}

/* end https://en.wikipedia.org/ */

</style></td></tr></tbody></table>

<table class="sidebar nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on <a href="Statistics" title="Statistics">statistics</a></td></tr><tr><th class="sidebar-title-with-pretitle"><a href="Probability_theory" title="Probability theory">Probability theory</a></th></tr><tr><td class="sidebar-image"><span class="skin-invert" typeof="mw:File"></span></td></tr><tr><td class="sidebar-content">

<ul><li><a href="Probability_axioms" title="Probability axioms">Axioms</a></li></ul></li>

<li><a href="Determinism" title="Determinism">Determinism</a>

<ul><li><a href="Deterministic_system" title="Deterministic system">System</a></li></ul></li>

<li><a href="Indeterminism" title="Indeterminism">Indeterminism</a></li>

<li><a href="Randomness" title="Randomness">Randomness</a></li></ul></td>

</tr><tr><td class="sidebar-content">

<ul><li><a href="Probability_space" title="Probability space">Probability space</a></li>

<li><a href="Sample_space" title="Sample space">Sample space</a></li>

<li><a href="Event_(probability_theory)" title="Event (probability theory)">Event</a>

<ul><li><a href="Collectively_exhaustive_events" title="Collectively exhaustive events">Collectively exhaustive events</a></li>

<li><a href="Elementary_event" title="Elementary event">Elementary event</a></li>

<li><a href="Mutual_exclusivity" title="Mutual exclusivity">Mutual exclusivity</a></li>

<li><a href="Outcome_(probability)" title="Outcome (probability)">Outcome</a></li>

<li><a href="Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li></ul></li>

<li><a href="Experiment_(probability_theory)" title="Experiment (probability theory)">Experiment</a>

<ul><li><a href="Bernoulli_trial" title="Bernoulli trial">Bernoulli trial</a></li></ul></li>

<li><a href="Probability_distribution" title="Probability distribution">Probability distribution</a>

<ul><li><a href="Bernoulli_distribution" title="Bernoulli distribution">Bernoulli distribution</a></li>

<li><a href="Binomial_distribution" title="Binomial distribution">Binomial distribution</a></li>

<li><a href="Exponential_distribution" title="Exponential distribution">Exponential distribution</a></li>

<li><a href="Normal_distribution" title="Normal distribution">Normal distribution</a></li>

<li><a href="Pareto_distribution" title="Pareto distribution">Pareto distribution</a></li>

<li><a href="Poisson_distribution" title="Poisson distribution">Poisson distribution</a></li></ul></li>

<li><a href="Probability_measure" title="Probability measure">Probability measure</a></li>

<li><a href="Random_variable" title="Random variable">Random variable</a>

<ul><li><a href="Bernoulli_process" title="Bernoulli process">Bernoulli process</a></li>

<li><a href="Continuous_or_discrete_variable" title="Continuous or discrete variable">Continuous or discrete</a></li>

<li><a href="Expected_value" title="Expected value">Expected value</a></li>

<li><a href="Variance" title="Variance">Variance</a></li>

<li><a href="Markov_chain" title="Markov chain">Markov chain</a></li>

<li><a href="Realization_(probability)" title="Realization (probability)">Observed value</a></li>

<li><a href="Random_walk" title="Random walk">Random walk</a></li>

<li><a href="Stochastic_process" title="Stochastic process">Stochastic process</a></li></ul></li></ul></td>

</tr><tr><td class="sidebar-content">

<ul><li><a href="Complementary_event" title="Complementary event">Complementary event</a></li>

<li><a href="Joint_probability_distribution" title="Joint probability distribution">Joint probability</a></li>

<li><a href="Marginal_distribution" title="Marginal distribution">Marginal probability</a></li>

<li><a href="Conditional_probability" title="Conditional probability">Conditional probability</a></li></ul></td>

</tr><tr><td class="sidebar-content">

<ul><li><a href="Independence_(probability_theory)" title="Independence (probability theory)">Independence</a></li>

<li><a href="Conditional_independence" title="Conditional independence">Conditional independence</a></li>

<li><a href="Law_of_total_probability" title="Law of total probability">Law of total probability</a></li>

<li><a href="Law_of_large_numbers" title="Law of large numbers">Law of large numbers</a></li>

<li><a href="Bayes'_theorem" title="Bayes' theorem">Bayes' theorem</a></li>

<li><a href="Boole's_inequality" title="Boole's inequality">Boole's inequality</a></li></ul></td>

</tr><tr><td class="sidebar-content">

<ul><li><a href="Venn_diagram" title="Venn diagram">Venn diagram</a></li>

<li><a href="Tree_diagram_(probability_theory)" title="Tree diagram (probability theory)">Tree diagram</a></li></ul></td>

</tr><tr><td class="sidebar-navbar"></td></tr></tbody></table>

<table class="sidebar nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="Mathematics" title="Mathematics">Mathematics</a></th></tr><tr><td class="sidebar-above" style="padding-bottom:0.35em;">

<ul><li><a href="History_of_mathematics" title="History of mathematics">History</a></li>

<li><a href="Lists_of_mathematics_topics" title="Lists of mathematics topics">Index</a></li></ul></td></tr><tr><td class="sidebar-content-with-subgroup">

<div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="border-top:1px solid #aaa;background:#ddddff;text-align:center;;color: var(--color-base)"><a href="Areas_of_mathematics" class="mw-redirect" title="Areas of mathematics">Areas</a></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Number_theory" title="Number theory">Number theory</a></li>

<li><a href="Geometry" title="Geometry">Geometry</a></li>

<li><a href="Algebra" title="Algebra">Algebra</a></li>

<li><a href="Calculus" title="Calculus">Calculus</a> and <a href="Mathematical_analysis" title="Mathematical analysis">Analysis</a></li>

<li><a href="Discrete_mathematics" title="Discrete mathematics">Discrete mathematics</a></li>

<li><a href="Mathematical_logic" title="Mathematical logic">Logic</a></li>

<li><a href="Set_theory" title="Set theory">Set theory</a></li>

<li><a href="Statistics" title="Statistics">Statistics</a> and <a href="Decision_theory" title="Decision theory">Decision theory</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="border-top:1px solid #aaa;background:#ddddff;text-align:center;;color: var(--color-base)">Relationship with sciences</div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Mathematical_physics" title="Mathematical physics">Physics</a></li>

<li><a href="Mathematical_chemistry" title="Mathematical chemistry">Chemistry</a></li>

<li><a href="Geomathematics" title="Geomathematics">Geosciences</a></li>

<li><a href="Computational_mathematics" title="Computational mathematics">Computation</a></li>

<li><a href="Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Biology</a></li>

<li><a href="Mathematical_linguistics" title="Mathematical linguistics">Linguistics</a></li>

<li><a href="Mathematical_economics" title="Mathematical economics">Economics</a></li>

<li><a href="Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy</a></li>

<li><a href="Mathematics_education" title="Mathematics education">Education</a></li></ul></div></div></td>

</tr></tbody></table></td>

</tr><tr><th class="sidebar-heading">

<span typeof="mw:File"></span> <a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics Portal</a></th></tr><tr><td class="sidebar-navbar"></td></tr></tbody></table>

<p><b>Probability</b> is a branch of <a href="Mathematics" title="Mathematics">mathematics</a> and <a href="Statistics" title="Statistics">statistics</a> concerning <a href="Event_(probability_theory)" title="Event (probability theory)">events</a> and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Stuart_and_Ord_2009_2-0" class="reference"><a href="#cite_note-Stuart_and_Ord_2009-2"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Feller_3-0" class="reference"><a href="#cite_note-Feller-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> This number is often expressed as a percentage (%), ranging from 0% to 100%. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%).

</p><p>These concepts have been given an <a href="Probability_axioms" title="Probability axioms">axiomatic</a> mathematical formalization in <i><a href="Probability_theory" title="Probability theory">probability theory</a></i>, which is used widely in <a href="Areas_of_study" class="mw-redirect" title="Areas of study">areas of study</a> such as <a href="Statistics" title="Statistics">statistics</a>, <a href="Mathematics" title="Mathematics">mathematics</a>, <a href="Science" title="Science">science</a>, <a href="Finance" title="Finance">finance</a>, <a href="Gambling" title="Gambling">gambling</a>, <a href="Artificial_intelligence" title="Artificial intelligence">artificial intelligence</a>, <a href="Machine_learning" title="Machine learning">machine learning</a>, <a href="Computer_science" title="Computer science">computer science</a>, <a href="Game_theory" title="Game theory">game theory</a>, and <a href="Philosophy" title="Philosophy">philosophy</a> to, for example, draw inferences about the expected frequency of events. Probability theory is also used to describe the underlying mechanics and regularities of <a href="Complex_systems" class="mw-redirect" title="Complex systems">complex systems</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Etymology">Etymology</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="History_of_probability#Etymology" title="History of probability">History of probability § Etymology</a>, and <a href="Glossary_of_probability_and_statistics" title="Glossary of probability and statistics">Glossary of probability and statistics</a></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Likelihood" class="mw-redirect" title="Likelihood">Likelihood</a></div>

<p>The word <i>probability</i> <a href="Etymology" title="Etymology">derives</a> from the Latin <span title="Latin-language text"><i lang="la">probabilitas</i></span>, which can also mean "<a href="https://en.wiktionary.org/wiki/probity" class="extiw external" title="wiktionary:probity">probity</a>", a measure of the <a href="Authority" title="Authority">authority</a> of a <a href="Witness" title="Witness">witness</a> in a <a href="Legal_case" class="mw-redirect" title="Legal case">legal case</a> in Europe, and often correlated with the witness's <a href="Nobility" title="Nobility">nobility</a>. In a sense, this differs much from the modern meaning of <i>probability</i>, which in contrast is a measure of the weight of <a href="Empirical_evidence" title="Empirical evidence">empirical evidence</a>, and is arrived at from <a href="Inductive_reasoning" title="Inductive reasoning">inductive reasoning</a> and <a href="Statistical_inference" title="Statistical inference">statistical inference</a>.<sup id="cite_ref-Emergence_5-0" class="reference"><a href="#cite_note-Emergence-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Interpretations">Interpretations</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Probability_interpretations" title="Probability interpretations">Probability interpretations</a></div>

<p>When dealing with <a href="Experiment_(probability_theory)" title="Experiment (probability theory)">random experiments</a> – i.e., <a href="Experiment" title="Experiment">experiments</a> that are <a href="Randomness" title="Randomness">random</a> and <a href="Well-defined_expression" title="Well-defined expression">well-defined</a> – in a purely theoretical setting (like tossing a coin), probabilities can be numerically described by the number of desired outcomes, divided by the total number of all outcomes. This is referred to as <b>theoretical probability</b> (in contrast to <a href="Empirical_probability" title="Empirical probability">empirical probability</a>, dealing with probabilities in the context of real experiments). The probability is a number between 0 and 1; the larger the probability, the more likely the desired outcome is to occur. For example, tossing a coin twice will yield "head-head", "head-tail", "tail-head", and "tail-tail" outcomes. The probability of getting an outcome of "head-head" is 1 out of 4 outcomes, or, in numerical terms, 1/4, 0.25 or 25%. The probability of getting an outcome of at least one head is 3 out of 4, or 0.75, and this event is more likely to occur. However, when it comes to practical application, there are two major competing categories of probability interpretations, whose adherents hold different views about the fundamental nature of probability:

</p>

<ul><li><a href="Objectivity_(philosophy)" class="mw-redirect" title="Objectivity (philosophy)">Objectivists</a> assign numbers to describe some objective or physical state of affairs. The most popular version of objective probability is <a href="Frequentist_probability" title="Frequentist probability">frequentist probability</a>, which claims that the probability of a random event denotes the <i><a href="Frequency_(statistics)" title="Frequency (statistics)">relative frequency</a> of occurrence</i> of an experiment's outcome when the experiment is repeated indefinitely. This interpretation considers probability to be the relative frequency "in the long run" of outcomes.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> A modification of this is <a href="Propensity_probability" title="Propensity probability">propensity probability</a>, which interprets probability as the tendency of some experiment to yield a certain outcome, even if it is performed only once.</li>

<li><a href="Subjective_probability" class="mw-redirect" title="Subjective probability">Subjectivists</a> assign numbers per subjective probability, that is, as a <a href="Credence_(statistics)" title="Credence (statistics)">degree of belief</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The degree of belief has been interpreted as "the price at which you would buy or sell a bet that pays 1 unit of utility if E, 0 if not E",<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> although that interpretation is not universally agreed upon.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The most popular version of subjective probability is <a href="Bayesian_probability" title="Bayesian probability">Bayesian probability</a>, which includes expert knowledge as well as experimental data to produce probabilities. The expert knowledge is represented by some (subjective) <a href="Prior_probability_distribution" class="mw-redirect" title="Prior probability distribution">prior probability distribution</a>. These data are incorporated in a <a href="Likelihood_function" title="Likelihood function">likelihood function</a>. The product of the prior and the likelihood, when normalized, results in a <a href="Posterior_probability_distribution" class="mw-redirect" title="Posterior probability distribution">posterior probability distribution</a> that incorporates all the information known to date.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> By <a href="Aumann's_agreement_theorem" title="Aumann's agreement theorem">Aumann's agreement theorem</a>, Bayesian agents whose prior beliefs are similar will end up with similar posterior beliefs. However, sufficiently different priors can lead to different conclusions, regardless of how much information the agents share.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></li></ul>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="History_of_probability" title="History of probability">History of probability</a></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="History_of_statistics" title="History of statistics">History of statistics</a></div>

<p>The scientific study of probability is a modern development of mathematics. <a href="Gambling" title="Gambling">Gambling</a> shows that there has been an interest in quantifying the ideas of probability throughout history, but exact mathematical descriptions arose much later. There are reasons for the slow development of the mathematics of probability. Whereas games of chance provided the impetus for the mathematical study of probability, fundamental issues <sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>note 2<span class="cite-bracket">]</span></a></sup> are still obscured by superstitions.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>

</p><p>According to <a href="Richard_Jeffrey" title="Richard Jeffrey">Richard Jeffrey</a>, "Before the middle of the seventeenth century, the term 'probable' (Latin <i>probabilis</i>) meant <i>approvable</i>, and was applied in that sense, univocally, to opinion and to action. A probable action or opinion was one such as sensible people would undertake or hold, in the circumstances."<sup id="cite_ref-Jeffrey_14-0" class="reference"><a href="#cite_note-Jeffrey-14"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> However, in legal contexts especially, 'probable' could also apply to propositions for which there was good evidence.<sup id="cite_ref-Franklin_15-0" class="reference"><a href="#cite_note-Franklin-15"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>

</p>

<p>The sixteenth-century <a href="Italians" title="Italians">Italian</a> polymath <a href="Gerolamo_Cardano" title="Gerolamo Cardano">Gerolamo Cardano</a> demonstrated the efficacy of defining <a href="Odds" title="Odds">odds</a> as the ratio of favourable to unfavourable outcomes (which implies that the probability of an event is given by the ratio of favourable outcomes to the total number of possible outcomes<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup>).

Aside from the elementary work by Cardano, the doctrine of probabilities dates to the correspondence of <a href="Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a> and <a href="Blaise_Pascal" title="Blaise Pascal">Blaise Pascal</a> (1654). <a href="Christiaan_Huygens" title="Christiaan Huygens">Christiaan Huygens</a> (1657) gave the earliest known scientific treatment of the subject.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> <a href="Jakob_Bernoulli" class="mw-redirect" title="Jakob Bernoulli">Jakob Bernoulli</a>'s <i><a href="Ars_Conjectandi" title="Ars Conjectandi">Ars Conjectandi</a></i> (posthumous, 1713) and <a href="Abraham_de_Moivre" title="Abraham de Moivre">Abraham de Moivre</a>'s <i><a href="The_Doctrine_of_Chances" title="The Doctrine of Chances">Doctrine of Chances</a></i> (1718) treated the subject as a branch of mathematics.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> See <a href="Ian_Hacking" title="Ian Hacking">Ian Hacking</a>'s <i>The Emergence of Probability</i><sup id="cite_ref-Emergence_5-1" class="reference"><a href="#cite_note-Emergence-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> and <a href="James_Franklin_(philosopher)" title="James Franklin (philosopher)">James Franklin's</a> <i>The Science of Conjecture</i><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> for histories of the early development of the very concept of mathematical probability.

</p><p>The <a href="Theory_of_errors" class="mw-redirect" title="Theory of errors">theory of errors</a> may be traced back to <a href="Roger_Cotes" title="Roger Cotes">Roger Cotes</a>'s <i>Opera Miscellanea</i> (posthumous, 1722), but a memoir prepared by <a href="Thomas_Simpson" title="Thomas Simpson">Thomas Simpson</a> in 1755 (printed 1756) first applied the theory to the discussion of errors of observation.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> The reprint (1757) of this memoir lays down the axioms that positive and negative errors are equally probable, and that certain assignable limits define the range of all errors. Simpson also discusses continuous errors and describes a probability curve.

</p><p>The first two laws of error that were proposed both originated with <a href="Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a>. The first law was published in 1774, and stated that the frequency of an error could be expressed as an exponential function of the numerical magnitude of the error – disregarding sign. The second law of error was proposed in 1778 by Laplace, and stated that the frequency of the error is an exponential function of the square of the error.<sup id="cite_ref-Wilson1923_21-0" class="reference"><a href="#cite_note-Wilson1923-21"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> The second law of error is called the normal distribution or the Gauss law. "It is difficult historically to attribute that law to Gauss, who in spite of his well-known precocity had probably not made this discovery before he was two years old."<sup id="cite_ref-Wilson1923_21-1" class="reference"><a href="#cite_note-Wilson1923-21"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup>

</p><p><a href="Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a> (1778) introduced the principle of the maximum product of the probabilities of a system of concurrent errors.

</p>

<p><a href="Adrien-Marie_Legendre" title="Adrien-Marie Legendre">Adrien-Marie Legendre</a> (1805) developed the <a href="Method_of_least_squares" class="mw-redirect" title="Method of least squares">method of least squares</a>, and introduced it in his <i>Nouvelles méthodes pour la détermination des orbites des comètes</i> (<i>New Methods for Determining the Orbits of Comets</i>).<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> In ignorance of Legendre's contribution, an Irish-American writer, <a href="Robert_Adrain" title="Robert Adrain">Robert Adrain</a>, editor of "The Analyst" (1808), first deduced the law of facility of error,

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (x)=ce^{-h^{2}x^{2}}}">

<msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi (x)=ce^{-h^{2}x^{2}}}</annotation>

</semantics>

</math></span></span>

</p><p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

</semantics>

</math></span><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"></span> is a constant depending on precision of observation, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

</semantics>

</math></span><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"></span> is a scale factor ensuring that the area under the curve equals 1. He gave two proofs, the second being essentially the same as <a href="John_Herschel" title="John Herschel">John Herschel</a>'s (1850). <a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a> gave the first proof that seems to have been known in Europe (the third after Adrain's) in 1809. Further proofs were given by Laplace (1810, 1812), Gauss (1823), <a href="James_Ivory_(mathematician)" title="James Ivory (mathematician)">James Ivory</a> (1825, 1826), Hagen (1837), <a href="Friedrich_Bessel" class="mw-redirect" title="Friedrich Bessel">Friedrich Bessel</a> (1838), <a href="William_Fishburn_Donkin" title="William Fishburn Donkin">W.F. Donkin</a> (1844, 1856), and <a href="Morgan_Crofton" title="Morgan Crofton">Morgan Crofton</a> (1870). Other contributors were <a href="Robert_Leslie_Ellis" title="Robert Leslie Ellis">Ellis</a> (1844), <a href="Augustus_De_Morgan" title="Augustus De Morgan">De Morgan</a> (1864), <a href="James_Whitbread_Lee_Glaisher" title="James Whitbread Lee Glaisher">Glaisher</a> (1872), and <a href="Giovanni_Schiaparelli" title="Giovanni Schiaparelli">Giovanni Schiaparelli</a> (1875). <a href="Christian_August_Friedrich_Peters" title="Christian August Friedrich Peters">Peters</a>'s (1856) formula for <i>r</i>, the <a href="Probable_error" title="Probable error">probable error</a> of a single observation, is well known.

</p><p>In the nineteenth century, authors on the general theory included <a href="Laplace" class="mw-redirect" title="Laplace">Laplace</a>, <a href="Sylvestre_Lacroix" class="mw-redirect" title="Sylvestre Lacroix">Sylvestre Lacroix</a> (1816), Littrow (1833), <a href="Adolphe_Quetelet" title="Adolphe Quetelet">Adolphe Quetelet</a> (1853), <a href="Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a> (1860), Helmert (1872), <a href="Hermann_Laurent" class="mw-redirect" title="Hermann Laurent">Hermann Laurent</a> (1873), Liagre, Didion and <a href="Karl_Pearson" title="Karl Pearson">Karl Pearson</a>. <a href="Augustus_De_Morgan" title="Augustus De Morgan">Augustus De Morgan</a> and <a href="George_Boole" title="George Boole">George Boole</a> improved the exposition of the theory.

</p><p>In 1906, <a href="Andrey_Markov" title="Andrey Markov">Andrey Markov</a> introduced<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> the notion of <a href="Markov_chains" class="mw-redirect" title="Markov chains">Markov chains</a>, which played an important role in <a href="Stochastic_process" title="Stochastic process">stochastic processes</a> theory and its applications. The modern theory of probability based on <a href="Measure_(mathematics)" title="Measure (mathematics)">measure theory</a> was developed by <a href="Andrey_Kolmogorov" title="Andrey Kolmogorov">Andrey Kolmogorov</a> in 1931.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup>

</p><p>On the geometric side, contributors to <i>The Educational Times</i> included Miller, Crofton, McColl, Wolstenholme, Watson, and <a href="Artemas_Martin" title="Artemas Martin">Artemas Martin</a>.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> See <a href="Integral_geometry" title="Integral geometry">integral geometry</a> for more information.

</p>

<div class="mw-heading mw-heading2"><h2 id="Theory">Theory</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Probability_theory" title="Probability theory">Probability theory</a></div>

<p>Like other <a href="Theory" title="Theory">theories</a>, the <a href="Probability_theory" title="Probability theory">theory of probability</a> is a representation of its concepts in formal terms – that is, in terms that can be considered separately from their meaning. These formal terms are manipulated by the rules of mathematics and logic, and any results are interpreted or translated back into the problem domain.

</p><p>There have been at least two successful attempts to formalize probability, namely the <a href="Kolmogorov" class="mw-redirect" title="Kolmogorov">Kolmogorov</a> formulation and the <a href="Richard_Threlkeld_Cox" title="Richard Threlkeld Cox">Cox</a> formulation. In Kolmogorov's formulation (see also <a href="Probability_space" title="Probability space">probability space</a>), <a href="Set_(mathematics)" title="Set (mathematics)">sets</a> are interpreted as <a href="Event_(probability_theory)" title="Event (probability theory)">events</a> and probability as a <a href="Measure_(mathematics)" title="Measure (mathematics)">measure</a> on a class of sets. In <a href="Cox's_theorem" title="Cox's theorem">Cox's theorem</a>, probability is taken as a primitive (i.e., not further analyzed), and the emphasis is on constructing a consistent assignment of probability values to propositions. In both cases, the <a href="Probability_axioms" title="Probability axioms">laws of probability</a> are the same, except for technical details.

</p><p>There are other methods for quantifying uncertainty, such as the <a href="Dempster%E2%80%93Shafer_theory" title="Dempster–Shafer theory">Dempster–Shafer theory</a> or <a href="Possibility_theory" title="Possibility theory">possibility theory</a>, but those are essentially different and not compatible with the usually-understood laws of probability.

</p>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<p>Probability theory is applied in everyday life in <a href="Risk" title="Risk">risk</a> assessment and <a href="Statistical_model" title="Statistical model">modeling</a>. The insurance industry and <a href="Market_(economics)" title="Market (economics)">markets</a> use <a href="Actuarial_science" title="Actuarial science">actuarial science</a> to determine pricing and make trading decisions. Governments apply probabilistic methods in <a href="Environmental_regulation" class="mw-redirect" title="Environmental regulation">environmental regulation</a>, entitlement analysis, and <a href="Financial_regulation" title="Financial regulation">financial regulation</a>.

</p><p>An example of the use of probability theory in equity trading is the effect of the perceived probability of any widespread Middle East conflict on oil prices, which have ripple effects in the economy as a whole. An assessment by a commodity trader that a war is more likely can send that commodity's prices up or down, and signals other traders of that opinion. Accordingly, the probabilities are neither assessed independently nor necessarily rationally. The theory of <a href="Behavioral_finance" class="mw-redirect" title="Behavioral finance">behavioral finance</a> emerged to describe the effect of such <a href="Groupthink" title="Groupthink">groupthink</a> on pricing, on policy, and on peace and conflict.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup>

</p><p>In addition to financial assessment, probability can be used to analyze trends in biology (e.g., disease spread) as well as ecology (e.g., biological <a href="Punnett_squares" class="mw-redirect" title="Punnett squares">Punnett squares</a>).<sup id="cite_ref-Edwards_2012_2_27-0" class="reference"><a href="#cite_note-Edwards_2012_2-27"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> As with finance, risk assessment can be used as a statistical tool to calculate the likelihood of undesirable events occurring, and can assist with implementing protocols to avoid encountering such circumstances. Probability is used to design <a href="Games_of_chance" class="mw-redirect" title="Games of chance">games of chance</a> so that casinos can make a guaranteed profit, yet provide payouts to players that are frequent enough to encourage continued play.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup>

</p><p>Another significant application of probability theory in everyday life is <a href="Reliability_(statistics)" title="Reliability (statistics)">reliability</a>. Many consumer products, such as <a href="Automobiles" class="mw-redirect" title="Automobiles">automobiles</a> and consumer electronics, use reliability theory in product design to reduce the probability of failure. Failure probability may influence a manufacturer's decisions on a product's <a href="Warranty" title="Warranty">warranty</a>.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup>

</p><p>The <a href="Cache_language_model" title="Cache language model">cache language model</a> and other <a href="Statistical_Language_Model" class="mw-redirect" title="Statistical Language Model">statistical language models</a> that are used in <a href="Natural_language_processing" title="Natural language processing">natural language processing</a> are also examples of applications of probability theory.

</p>

<div class="mw-heading mw-heading2"><h2 id="Mathematical_treatment">Mathematical treatment</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Probability_axioms" title="Probability axioms">Probability axioms</a></div>

<p>Consider an experiment that can produce a number of results. The collection of all possible results is called the <a href="Sample_space" title="Sample space">sample space</a> of the experiment, sometimes denoted as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span>. The <a href="Power_set" title="Power set">power set</a> of the sample space is formed by considering all different collections of possible results. For example, rolling a die can produce six possible results. One collection of possible results gives an odd number on the die. Thus, the subset {1,3,5} is an element of the <a href="Power_set" title="Power set">power set</a> of the sample space of dice rolls. These collections are called "events". In this case, {1,3,5} is the event that the die falls on some odd number. If the results that actually occur fall in a given event, the event is said to have occurred.

</p><p>A probability is a <a href="Function_(mathematics)" title="Function (mathematics)">way of assigning</a> every event a value between zero and one, with the requirement that the event made up of all possible results (in our example, the event {1,2,3,4,5,6}) is assigned a value of one. To qualify as a probability, the assignment of values must satisfy the requirement that for any collection of mutually exclusive events (events with no common results, such as the events {1,6}, {3}, and {2,4}), the probability that at least one of the events will occur is given by the sum of the probabilities of all the individual events.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup>

</p><p>The probability of an <a href="Event_(probability_theory)" title="Event (probability theory)">event</a> <i>A</i> is written as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(A)}</annotation>

</semantics>

</math></span><img src="./4f264d19e21604793c6dc54f8044df454db82744.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.298ex; height:2.843ex;" alt="{\displaystyle P(A)}" loading="lazy"></span>,<sup id="cite_ref-:2_31-0" class="reference"><a href="#cite_note-:2-31"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(A)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(A)}</annotation>

</semantics>

</math></span><img src="./5198674703aa51158cfb468492d0daf78b96bb4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.811ex; height:2.843ex;" alt="{\displaystyle p(A)}" loading="lazy"></span>, or <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Pr}}(A)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Pr}}(A)}</annotation>

</semantics>

</math></span><img src="./837f66c6989a2f8aa481bc156255fc83da5290f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.047ex; height:2.843ex;" alt="{\displaystyle {\text{Pr}}(A)}" loading="lazy"></span>.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> This mathematical definition of probability can extend to infinite sample spaces, and even uncountable sample spaces, using the concept of a measure.

</p><p>The <i>opposite</i> or <i>complement</i> of an event <i>A</i> is the event [not <i>A</i>] (that is, the event of <i>A</i> not occurring), often denoted as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A',A^{c}}">

<msup>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A',A^{c}}</annotation>

</semantics>

</math></span><img src="./e7c4e0d4816cfbc368fc7f4eca85c948ed369f8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.149ex; height:2.843ex;" alt="{\displaystyle A',A^{c}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {A}},A^{\complement },\neg A}">

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {A}},A^{\complement },\neg A}</annotation>

</semantics>

</math></span><img src="./09a26a0a613647e207f0ed7e6ce972589c5406a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.017ex; height:3.343ex;" alt="{\displaystyle {\overline {A}},A^{\complement },\neg A}" loading="lazy"></span>, or <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sim }A}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sim }A}</annotation>

</semantics>

</math></span><img src="./b68eaec09624f5807084ecbda6a27f4ba88b6d2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.551ex; height:2.176ex;" alt="{\displaystyle {\sim }A}" loading="lazy"></span>; its probability is given by <span class="nowrap"><i>P</i>(not <i>A</i>) = 1 − <i>P</i>(<i>A</i>)</span>.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> As an example, the chance of not rolling a six on a six-sided die is <span class="nowrap">1 – (chance of rolling a six) =</span> <span class="nowrap">1 − <style data-mw-deduplicate="TemplateStyles:r1214402035">

/* start https://en.wikipedia.org/ */

.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}

/* end https://en.wikipedia.org/ */

</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">6</span></span>⁠</span> = <span class="sfrac">⁠<span class="tion"><span class="num">5</span><span class="sr-only">/</span><span class="den">6</span></span>⁠</span>.</span> For a more comprehensive treatment, see <a href="Complementary_event" title="Complementary event">Complementary event</a>.

</p><p>If two events <i>A</i> and <i>B</i> occur on a single performance of an experiment, this is called the intersection or <a href="Joint_distribution" class="mw-redirect" title="Joint distribution">joint probability</a> of <i>A</i> and <i>B</i>, denoted as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A\cap B).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(A\cap B).}</annotation>

</semantics>

</math></span><img src="./79a0bb6aa1a0a1dc3da6f8307cd9bbac5913920a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.291ex; height:2.843ex;" alt="{\displaystyle P(A\cap B).}" loading="lazy"></span>

</p>

<div class="mw-heading mw-heading3"><h3 id="Independent_events">Independent events</h3></div>

<p>If two events, <i>A</i> and <i>B</i> are <a href="Independence_(probability_theory)" title="Independence (probability theory)">independent</a> then the joint probability is<sup id="cite_ref-:2_31-1" class="reference"><a href="#cite_note-:2-31"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup>

</p><p><span class="mwe-math-element mwe-math-element-block" data-qid="Q120632573"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A{\mbox{ and }}B)=P(A\cap B)=P(A)P(B).}">

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(A{\mbox{ and }}B)=P(A\cap B)=P(A)P(B).}</annotation>

</semantics>

</math></span></span>

</p>

<p>For example, if two coins are flipped, then the chance of both being heads is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}\times {\tfrac {1}{2}}={\tfrac {1}{4}}.}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}\times {\tfrac {1}{2}}={\tfrac {1}{4}}.}</annotation>

</semantics>

</math></span><img src="./a6308df32ec2d2dda8eab361f11a36e5241b0077.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.56ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}\times {\tfrac {1}{2}}={\tfrac {1}{4}}.}" loading="lazy"></span><sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Mutually_exclusive_events">Mutually exclusive events</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Mutual_exclusivity" title="Mutual exclusivity">Mutual exclusivity</a></div>

<p>If either event <i>A</i> or event <i>B</i> can occur but never both simultaneously, then they are called mutually exclusive events.

</p><p>If two events are <a href="Mutually_exclusive_events" class="mw-redirect" title="Mutually exclusive events">mutually exclusive</a>, then the probability of <i>both</i> occurring is denoted as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A\cap B)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(A\cap B)}</annotation>

</semantics>

</math></span><img src="./f22276bc48d131dadc7e4dacbf38cee3ed05d536.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.644ex; height:2.843ex;" alt="{\displaystyle P(A\cap B)}" loading="lazy"></span> and<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A{\mbox{ and }}B)=P(A\cap B)=0}">

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(A{\mbox{ and }}B)=P(A\cap B)=0}</annotation>

</semantics>

</math></span></span> If two events are <a href="Mutually_exclusive_events" class="mw-redirect" title="Mutually exclusive events">mutually exclusive</a>, then the probability of <i>either</i> occurring is denoted as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A\cup B)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(A\cup B)}</annotation>

</semantics>

</math></span><img src="./c6a6862183808b257377dc7be2a2716ccbd70142.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.644ex; height:2.843ex;" alt="{\displaystyle P(A\cup B)}" loading="lazy"></span> and<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A{\mbox{ or }}B)=P(A\cup B)=P(A)+P(B)-P(A\cap B)=P(A)+P(B)-0=P(A)+P(B)}">

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(A{\mbox{ or }}B)=P(A\cup B)=P(A)+P(B)-P(A\cap B)=P(A)+P(B)-0=P(A)+P(B)}</annotation>

</semantics>

</math></span></span>

</p><p>For example, the chance of rolling a 1 or 2 on a six-sided die is <span class="mwe-math-element mwe-math-element-inline"><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{\mbox{ or }}2)=P(1)+P(2)={\tfrac {1}{6}}+{\tfrac {1}{6}}={\tfrac {1}{3}}.}">

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(1{\mbox{ or }}2)=P(1)+P(2)={\tfrac {1}{6}}+{\tfrac {1}{6}}={\tfrac {1}{3}}.}</annotation>

</semantics>

</math></span><img src="./f46e76d8c31d9d1eeca4e647b338bdfd1f69e785.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:39.147ex; height:3.676ex;" alt="{\displaystyle P(1{\mbox{ or }}2)=P(1)+P(2)={\tfrac {1}{6}}+{\tfrac {1}{6}}={\tfrac {1}{3}}.}" loading="lazy"></span>

</p>

<div class="mw-heading mw-heading3"><h3 id="Not_(necessarily)_mutually_exclusive_events">Not (necessarily) mutually exclusive events</h3></div>

<p>If the events are not (necessarily) mutually exclusive then<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\left(A{\hbox{ or }}B\right)=P(A\cup B)=P\left(A\right)+P\left(B\right)-P\left(A{\mbox{ and }}B\right).}">

<mrow>

<mrow>

</mstyle>

</mrow>

</mrow>

</mrow>

<mrow>

</mrow>

<mrow>

</mrow>

<mrow>

<mrow>

</mstyle>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P\left(A{\hbox{ or }}B\right)=P(A\cup B)=P\left(A\right)+P\left(B\right)-P\left(A{\mbox{ and }}B\right).}</annotation>

</semantics>

</math></span></span> Rewritten,<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)}">

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

<mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)}</annotation>

</semantics>

</math></span></span>

</p><p>For example, when drawing a card from a deck of cards, the chance of getting a heart or a face card (J, Q, K) (or both) is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {13}{52}}+{\tfrac {12}{52}}-{\tfrac {3}{52}}={\tfrac {11}{26}},}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {13}{52}}+{\tfrac {12}{52}}-{\tfrac {3}{52}}={\tfrac {11}{26}},}</annotation>

</semantics>

</math></span><img src="./580753566a427dabeafeda0cb3d80fee44cac8b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:19.346ex; height:3.676ex;" alt="{\displaystyle {\tfrac {13}{52}}+{\tfrac {12}{52}}-{\tfrac {3}{52}}={\tfrac {11}{26}},}" loading="lazy"></span> since among the 52 cards of a deck, 13 are hearts, 12 are face cards, and 3 are both: here the possibilities included in the "3 that are both" are included in each of the "13 hearts" and the "12 face cards", but should only be counted once.

</p><p>This can be expanded further for multiple not (necessarily) mutually exclusive events. For three events, this proceeds as follows:<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}P\left(A\cup B\cup C\right)=&P\left(\left(A\cup B\right)\cup C\right)\\=&P\left(A\cup B\right)+P\left(C\right)-P\left(\left(A\cup B\right)\cap C\right)\\=&P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)+P\left(C\right)-P\left(\left(A\cap C\right)\cup \left(B\cap C\right)\right)\\=&P\left(A\right)+P\left(B\right)+P\left(C\right)-P\left(A\cap B\right)-\left(P\left(A\cap C\right)+P\left(B\cap C\right)-P\left(\left(A\cap C\right)\cap \left(B\cap C\right)\right)\right)\\P\left(A\cup B\cup C\right)=&P\left(A\right)+P\left(B\right)+P\left(C\right)-P\left(A\cap B\right)-P\left(A\cap C\right)-P\left(B\cap C\right)+P\left(A\cap B\cap C\right)\end{aligned}}}">

<mtr>

<mtd>

<mrow>

<mrow>

</mrow>

</mrow>

</mtd>

<mtd>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mrow>

</mrow>

<mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mrow>

</mrow>

<mrow>

</mrow>

<mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mrow>

<mrow>

</mrow>

</mrow>

</mtd>

<mtd>

<mrow>

</mrow>

<mrow>

</mrow>

<mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P\left(A\cup B\cup C\right)=&P\left(\left(A\cup B\right)\cup C\right)\\=&P\left(A\cup B\right)+P\left(C\right)-P\left(\left(A\cup B\right)\cap C\right)\\=&P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)+P\left(C\right)-P\left(\left(A\cap C\right)\cup \left(B\cap C\right)\right)\\=&P\left(A\right)+P\left(B\right)+P\left(C\right)-P\left(A\cap B\right)-\left(P\left(A\cap C\right)+P\left(B\cap C\right)-P\left(\left(A\cap C\right)\cap \left(B\cap C\right)\right)\right)\\P\left(A\cup B\cup C\right)=&P\left(A\right)+P\left(B\right)+P\left(C\right)-P\left(A\cap B\right)-P\left(A\cap C\right)-P\left(B\cap C\right)+P\left(A\cap B\cap C\right)\end{aligned}}}</annotation>

</semantics>

</math></span></span>It can be seen, then, that this pattern can be repeated for any number of events.

</p>

<div class="mw-heading mw-heading3"><h3 id="Conditional_probability">Conditional probability</h3></div>

<p><i><a href="Conditional_probability" title="Conditional probability">Conditional probability</a></i> is the probability of some event <i>A</i>, given the occurrence of some other event <i>B</i>. Conditional probability is written <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A\mid B)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(A\mid B)}</annotation>

</semantics>

</math></span><img src="./8f8f30f4da85b53901e0871eb41ed8827f511bb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.999ex; height:2.843ex;" alt="{\displaystyle P(A\mid B)}" loading="lazy"></span>, and is read "the probability of <i>A</i>, given <i>B</i>". It is defined by<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup>

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}\,}">

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}\,}</annotation>

</semantics>

</math></span></span>

</p><p>If <span class="mwe-math-element mwe-math-element-inline"><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(B)=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(B)=0}</annotation>

</semantics>

</math></span><img src="./973532626488736dc14f2ab2add7c09c7b2132ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.58ex; height:2.843ex;" alt="{\displaystyle P(B)=0}" loading="lazy"></span> then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A\mid B)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(A\mid B)}</annotation>

</semantics>

</math></span><img src="./8f8f30f4da85b53901e0871eb41ed8827f511bb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.999ex; height:2.843ex;" alt="{\displaystyle P(A\mid B)}" loading="lazy"></span> is formally <a href="Undefined_(mathematics)" title="Undefined (mathematics)">undefined</a> by this expression. In this case <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math></span><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span> are independent, since <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A\cap B)=P(A)P(B)=0.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(A\cap B)=P(A)P(B)=0.}</annotation>

</semantics>

</math></span><img src="./f005396b1aa54e16626b9aae4927d9fcfe9a877d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.267ex; height:2.843ex;" alt="{\displaystyle P(A\cap B)=P(A)P(B)=0.}" loading="lazy"></span> However, it is possible to define a conditional probability for some zero-probability events, for example by using a <a href="%CE%A3-algebra" title="Σ-algebra">σ-algebra</a> of such events (such as those arising from a <a href="Continuous_random_variable" class="mw-redirect" title="Continuous random variable">continuous random variable</a>).<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup>

</p><p>For example, in a bag of 2 red balls and 2 blue balls (4 balls in total), the probability of taking a red ball is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/2;}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1/2;}</annotation>

</semantics>

</math></span><img src="./a7a9a3d85a28414e2126478f4479a0cddaa921ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle 1/2;}" loading="lazy"></span> however, when taking a second ball, the probability of it being either a red ball or a blue ball depends on the ball previously taken. For example, if a red ball was taken, then the probability of picking a red ball again would be <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/3,}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1/3,}</annotation>

</semantics>

</math></span><img src="./9f69bbe36ec86b266ddb879d8e8123c722574a18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle 1/3,}" loading="lazy"></span> since only 1 red and 2 blue balls would have been remaining. And if a blue ball was taken previously, the probability of taking a red ball will be <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2/3.}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2/3.}</annotation>

</semantics>

</math></span><img src="./953f1f8b7d655c1d1ab46aef4adc1ade9c06faff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle 2/3.}" loading="lazy"></span>

</p>

<div class="mw-heading mw-heading3"><h3 id="Inverse_probability">Inverse probability</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Inverse_probability" title="Inverse probability">Inverse probability</a></div>

<p>In <a href="Probability_theory" title="Probability theory">probability theory</a> and applications, <i><a href="Bayes'_theorem" title="Bayes' theorem">Bayes' rule</a></i> relates the <a href="Odds" title="Odds">odds</a> of event <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{1}}</annotation>

</semantics>

</math></span><img src="./6bc2435b217c1a0f46f8a517ffa225c6f9440e81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{1}}" loading="lazy"></span> to event <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2},}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{2},}</annotation>

</semantics>

</math></span><img src="./6a01dea03d426ee6e3efd7978055e7cce45ca779.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.444ex; height:2.509ex;" alt="{\displaystyle A_{2},}" loading="lazy"></span> before (prior to) and after (posterior to) <a href="Conditional_probability" title="Conditional probability">conditioning</a> on another event <span class="mwe-math-element mwe-math-element-inline"><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.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B.}</annotation>

</semantics>

</math></span><img src="./0eccf5bca7cdc1fa4439af2d31831db6bde00473.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.411ex; height:2.176ex;" alt="{\displaystyle B.}" loading="lazy"></span> The odds on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{1}}</annotation>

</semantics>

</math></span><img src="./6bc2435b217c1a0f46f8a517ffa225c6f9440e81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{1}}" loading="lazy"></span> to event <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{2}}</annotation>

</semantics>

</math></span><img src="./3ec73b8bc9abc3efb934f5a6ec2803713771f4bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{2}}" loading="lazy"></span> is simply the ratio of the probabilities of the two events. When arbitrarily many events <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> are of interest, not just two, the rule can be rephrased as <i>posterior is proportional to prior times likelihood</i>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A|B)\propto P(A)P(B|A)}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(A|B)\propto P(A)P(B|A)}</annotation>

</semantics>

</math></span><img src="./68d07547ef6cacd3521eabb735d80af39b466b90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.814ex; height:2.843ex;" alt="{\displaystyle P(A|B)\propto P(A)P(B|A)}" loading="lazy"></span> where the proportionality symbol means that the left hand side is proportional to (i.e., equals a constant times) the right hand side as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> varies, for fixed or given <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math></span><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span> (Lee, 2012; Bertsch McGrayne, 2012). In this form it goes back to Laplace (1774) and to Cournot (1843); see Fienberg (2005).

</p>

<div class="mw-heading mw-heading3"><h3 id="Summary_of_probabilities">Summary of probabilities</h3></div>

<caption>Summary of probabilities

</caption>

<th scope="col">Event

</th>

<th scope="col">Probability

</th></tr>

<tr>

<th scope="row" style="text-align: center;">A

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(A)\in [0,1]}</annotation>

</semantics>

</math></span><img src="./8f07aab47c06682ab4a8b4934f5db8c813e3eff6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.791ex; height:2.843ex;" alt="{\displaystyle P(A)\in [0,1]}" loading="lazy"></span>

</td></tr>

<tr>

<th scope="row" style="text-align: center;">not A

</th>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(A^{\complement })=1-P(A)\,}</annotation>

</semantics>

</math></span><img src="./ab27972b25cfc21f09a36a069f40b27a1d7d7b97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.138ex; height:3.509ex;" alt="{\displaystyle P(A^{\complement })=1-P(A)\,}" loading="lazy"></span>

</td></tr>

<tr>

<th scope="row" style="text-align: center;">A or B

</th>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mtext>if A and B are mutually exclusive</mtext>

</mstyle>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P(A\cup B)&=P(A)+P(B)-P(A\cap B)\\P(A\cup B)&=P(A)+P(B)\qquad {\mbox{if A and B are mutually exclusive}}\\\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./65c4d0c6db14685663e754e1dc7d0ce72572cf4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:64.846ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}P(A\cup B)&=P(A)+P(B)-P(A\cap B)\\P(A\cup B)&=P(A)+P(B)\qquad {\mbox{if A and B are mutually exclusive}}\\\end{aligned}}}" loading="lazy"></span>

</td></tr>

<tr>

<th scope="row" style="text-align: center;">A and B

</th>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mtext>if A and B are independent</mtext>

</mstyle>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P(A\cap B)&=P(A|B)P(B)=P(B|A)P(A)\\P(A\cap B)&=P(A)P(B)\qquad {\mbox{if A and B are independent}}\\\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./17dc5877448c0aa0c71b59f8af57252c7c3b8622.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:55.664ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}P(A\cap B)&=P(A|B)P(B)=P(B|A)P(A)\\P(A\cap B)&=P(A)P(B)\qquad {\mbox{if A and B are independent}}\\\end{aligned}}}" loading="lazy"></span>

</td></tr>

<tr>

<th scope="row" style="text-align: center;">A given B

</th>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}={\frac {P(B|A)P(A)}{P(B)}}\,}</annotation>

</semantics>

</math></span><img src="./7b08fd4f5b0007f40ccb2dbf77fc849b240e789e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.906ex; height:6.509ex;" alt="{\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}={\frac {P(B|A)P(A)}{P(B)}}\,}" loading="lazy"></span>

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Relation_to_randomness_and_probability_in_quantum_mechanics">Relation to randomness and probability in quantum mechanics</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Randomness" title="Randomness">Randomness</a></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Quantum_fluctuation#Interpretations" title="Quantum fluctuation">Quantum fluctuation § Interpretations</a></div>

<p>In a <a href="Determinism" title="Determinism">deterministic</a> universe, based on <a href="Newtonian_mechanics" class="mw-redirect" title="Newtonian mechanics">Newtonian</a> concepts, there would be no probability if all conditions were known (<a href="Laplace's_demon" title="Laplace's demon">Laplace's demon</a>) (but there are situations in which <a href="Chaos_theory" title="Chaos theory">sensitivity to initial conditions</a> exceeds our ability to measure them, i.e. know them). In the case of a <a href="Roulette" title="Roulette">roulette</a> wheel, if the force of the hand and the period of that force are known, the number on which the ball will stop would be a certainty (though as a practical matter, this would likely be true only of a roulette wheel that had not been exactly levelled – as Thomas A. Bass' <a href="Eudaemons" title="Eudaemons">Newtonian Casino</a> revealed). This also assumes knowledge of inertia and friction of the wheel, weight, smoothness, and roundness of the ball, variations in hand speed during the turning, and so forth. A probabilistic description can thus be more useful than Newtonian mechanics for analyzing the pattern of outcomes of repeated rolls of a roulette wheel. Physicists face the same situation in the <a href="Kinetic_theory_of_gases" title="Kinetic theory of gases">kinetic theory of gases</a>, where the system, while deterministic <i>in principle</i>, is so complex (with the number of molecules typically the order of magnitude of the <a href="Avogadro_constant" title="Avogadro constant">Avogadro constant</a> <span class="nowrap">6.02<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>23</sup></span>) that only a statistical description of its properties is feasible.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup>

</p><p><a href="Probability_theory" title="Probability theory">Probability theory</a> is required to describe quantum phenomena.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> A revolutionary discovery of early 20th century <a href="Physics" title="Physics">physics</a> was the random character of all physical processes that occur at sub-atomic scales and are governed by the laws of <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>. The objective <a href="Wave_function" title="Wave function">wave function</a> evolves deterministically but, according to the <a href="Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen interpretation</a>, it deals with probabilities of observing, the outcome being explained by a <a href="Wave_function_collapse" title="Wave function collapse">wave function collapse</a> when an observation is made. However, the loss of <a href="Determinism" title="Determinism">determinism</a> for the sake of <a href="Instrumentalism" title="Instrumentalism">instrumentalism</a> did not meet with universal approval. <a href="Albert_Einstein" title="Albert Einstein">Albert Einstein</a> famously <a href="https://de.wikipedia.org/wiki/Albert_Einstein#Quellenangaben_und_Anmerkungen" class="extiw external" title="de:Albert Einstein">remarked</a> in a letter to <a href="Max_Born" title="Max Born">Max Born</a>: "I am convinced that God does not play dice".<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> Like Einstein, <a href="Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Erwin Schrödinger</a>, who <a href="Schr%C3%B6dinger_equation#Historical_background_and_development" title="Schrödinger equation">discovered</a> the wave function, believed quantum mechanics is a <a href="Statistical" class="mw-redirect" title="Statistical">statistical</a> approximation of an underlying deterministic <a href="Reality" title="Reality">reality</a>.<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> In some modern interpretations of the statistical mechanics of measurement, <a href="Quantum_decoherence" title="Quantum decoherence">quantum decoherence</a> is invoked to account for the appearance of subjectively probabilistic experimental outcomes.

</p>

/* start https://en.wikipedia.org/ */

.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}

/* end https://en.wikipedia.org/ */

</style>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Outline_of_probability" title="Outline of probability">Outline of probability</a></div>

<ul><li><a href="Contingency_(philosophy)" title="Contingency (philosophy)">Contingency</a></li>

<li><a href="Equiprobability" title="Equiprobability">Equiprobability</a></li>

<li><a href="Fuzzy_logic" title="Fuzzy logic">Fuzzy logic</a></li>

<li><a href="Heuristic_(psychology)" title="Heuristic (psychology)">Heuristic (psychology)</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

/* start https://en.wikipedia.org/ */

.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}

/* end https://en.wikipedia.org/ */

</style><div class="reflist">

<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Strictly speaking, a probability of 0 indicates that an event <a href="Almost_surely" title="Almost surely"><i>almost</i> never</a> takes place, whereas a probability of 1 indicates than an event <a href="Almost_surely" title="Almost surely"><i>almost</i> certainly</a> takes place. This is an important distinction when the <a href="Sample_space" title="Sample space">sample space</a> is infinite. For example, for the <a href="Continuous_uniform_distribution" title="Continuous uniform distribution">continuous uniform distribution</a> on the <a href="Real_number" title="Real number">real</a> interval [5, 10], there are an infinite number of possible outcomes, and the probability of any given outcome being observed – for instance, exactly 7 – is 0. This means that an observation will <i>almost surely not</i> be exactly 7. However, it does <b>not</b> mean that exactly 7 is <i>impossible</i>. Ultimately some specific outcome (with probability 0) will be observed, and one possibility for that specific outcome is exactly 7.</span>

</li>

<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">In the context of the book that this is quoted from, it is the theory of probability and the logic behind it that governs the phenomena of such things compared to rash predictions that rely on pure luck or mythological arguments such as gods of luck helping the winner of the game.</span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<li id="cite_note-Stuart_and_Ord_2009-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Stuart_and_Ord_2009_2-0">^</a></b></span> <span class="reference-text">"Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th ed., (2009), <style data-mw-deduplicate="TemplateStyles:r1238218222">

/* start https://en.wikipedia.org/ */

.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("./mw/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("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/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}}

/* end https://en.wikipedia.org/ */

</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-534-24312-8</bdi>.</span>

</li>

<li id="cite_note-Feller-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Feller_3-0">^</a></b></span> <span class="reference-text">William Feller, <i>An Introduction to Probability Theory and Its Applications</i>, vol. 1, 3rd ed., (1968), Wiley, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-25708-7</bdi>.</span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.britannica.com/EBchecked/topic/477530/probability-theory">Probability Theory</a>. The Britannica website.</span>

</li>

<li id="cite_note-Emergence-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Emergence_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Emergence_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="Ian_Hacking" title="Ian Hacking">Hacking, I.</a> (2006) <i>The Emergence of Probability: A Philosophical Study of Early Ideas about Probability, Induction and Statistical Inference</i>, Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-68557-3</bdi> </span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFHacking1965" class="citation book cs1"><a href="Ian_Hacking" title="Ian Hacking">Hacking, Ian</a> (1965). <i>The Logic of Statistical Inference</i>. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-05165-1</bdi>.</cite></span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFFinetti1970" class="citation journal cs1">Finetti, Bruno de (1970). "Logical foundations and measurement of subjective probability". <i>Acta Psychologica</i>. <b>34</b>: <span class="nowrap">129–</span>145. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0001-6918%2870%2990012-0">10.1016/0001-6918(70)90012-0</a>.</cite></span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFHájek2002" class="citation journal cs1">Hájek, Alan (21 October 2002). Edward N. Zalta (ed.). <a rel="nofollow" class="external text" href="http://plato.stanford.edu/archives/win2012/entries/probability-interpret/">"Interpretations of Probability"</a>. <i>The Stanford Encyclopedia of Philosophy</i> (Winter 2012 ed.)<span class="reference-accessdate">. Retrieved <span class="nowrap">22 April</span> 2013</span>.</cite></span>

</li>

<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite id="CITEREFJaynes2003" class="citation book cs1">Jaynes, E.T. (2003). "Section A.2 The de Finetti system of probability". In Bretthorst, G. Larry (ed.). <i>Probability Theory: The Logic of Science</i> (1 ed.). Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-59271-0</bdi>.</cite></span>

</li>

<li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><cite id="CITEREFHoggCraigMcKean2004" class="citation book cs1">Hogg, Robert V.; Craig, Allen; McKean, Joseph W. (2004). <i>Introduction to Mathematical Statistics</i> (6th ed.). Upper Saddle River: Pearson. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-13-008507-8</bdi>.</cite></span>

</li>

<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><cite id="CITEREFJaynes2003" class="citation book cs1">Jaynes, E.T. (2003). "Section 5.3 Converging and diverging views". In Bretthorst, G. Larry (ed.). <i>Probability Theory: The Logic of Science</i> (1 ed.). Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-59271-0</bdi>.</cite></span>

</li>

<li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a href="John_E._Freund" title="John E. Freund">Freund, John.</a> (1973) <i>Introduction to Probability</i>. Dickenson <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8221-0078-2</bdi> (p. 1)</span>

</li>

<li id="cite_note-Jeffrey-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-Jeffrey_14-0">^</a></b></span> <span class="reference-text">Jeffrey, R.C., <i>Probability and the Art of Judgment,</i> Cambridge University Press. (1992). pp. 54–55 . <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-39459-7</bdi></span>

</li>

<li id="cite_note-Franklin-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-Franklin_15-0">^</a></b></span> <span class="reference-text">Franklin, J. (2001) <i>The Science of Conjecture: Evidence and Probability Before Pascal,</i> Johns Hopkins University Press. (pp. 22, 113, 127)</span>

</li>

<li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.columbia.edu/~pg2113/index_files/Gorroochurn-Some%20Laws.pdf">"<i>Some laws and problems in classical probability and how Cardano anticipated them</i> Gorrochum, P. <i>Chance</i> magazine 2012"</a> <span class="cs1-format">(PDF)</span>.</cite></span>

</li>

<li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><cite id="CITEREFAbrams" class="citation cs1">Abrams, William. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170724052656/http://www.secondmoment.org/articles/probability.php"><i>A Brief History of Probability</i></a>. Second Moment. Archived from <a rel="nofollow" class="external text" href="http://www.secondmoment.org/articles/probability.php">the original</a> on 24 July 2017<span class="reference-accessdate">. Retrieved <span class="nowrap">23 May</span> 2008</span>.</cite></span>

</li>

<li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><cite id="CITEREFIvancevicIvancevic2008" class="citation book cs1">Ivancevic, Vladimir G.; Ivancevic, Tijana T. (2008). <i>Quantum leap : from Dirac and Feynman, across the universe, to human body and mind</i>. Singapore; Hackensack, NJ: World Scientific. p. 16. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-981-281-927-7</bdi>.</cite></span>

</li>

<li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><cite id="CITEREFFranklin2001" class="citation book cs1">Franklin, James (2001). <i>The Science of Conjecture: Evidence and Probability Before Pascal</i>. Johns Hopkins University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8018-6569-5</bdi>.</cite></span>

</li>

<li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><cite id="CITEREFShoesmith1985" class="citation journal cs1">Shoesmith, Eddie (November 1985). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0315-0860%2885%2990044-8">"Thomas Simpson and the arithmetic mean"</a>. <i>Historia Mathematica</i>. <b>12</b> (4): <span class="nowrap">352–</span>355. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0315-0860%2885%2990044-8">10.1016/0315-0860(85)90044-8</a></span>.</cite></span>

</li>

<li id="cite_note-Wilson1923-21"><span class="mw-cite-backlink">^ <a href="#cite_ref-Wilson1923_21-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Wilson1923_21-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Wilson EB (1923) "First and second laws of error". <a href="Journal_of_the_American_Statistical_Association" title="Journal of the American Statistical Association">Journal of the American Statistical Association</a>, 18, 143</span>

</li>

<li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><cite id="CITEREFSeneta" class="citation web cs1">Seneta, Eugene William. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160203070724/http://statprob.com/encyclopedia/AdrienMarieLegendre.html">""Adrien-Marie Legendre" (version 9)"</a>. <i>StatProb: The Encyclopedia Sponsored by Statistics and Probability Societies</i>. Archived from <a rel="nofollow" class="external text" href="http://statprob.com/encyclopedia/AdrienMarieLegendre.html">the original</a> on 3 February 2016<span class="reference-accessdate">. Retrieved <span class="nowrap">27 January</span> 2016</span>.</cite></span>

</li>

<li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><cite id="CITEREFWeber" class="citation web cs1">Weber, Richard. <a rel="nofollow" class="external text" href="http://www.statslab.cam.ac.uk/~rrw1/markov/M.pdf">"Markov Chains"</a> <span class="cs1-format">(PDF)</span>. <i>Statistical Laboratory</i>. University of Cambridge.</cite></span>

</li>

<li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><cite id="CITEREFVitanyi1988" class="citation journal cs1">Vitanyi, Paul M.B. (1988). <a rel="nofollow" class="external text" href="http://homepages.cwi.nl/~paulv/KOLMOGOROV.BIOGRAPHY.html">"Andrei Nikolaevich Kolmogorov"</a>. <i>CWI Quarterly</i> (1): <span class="nowrap">3–</span>18<span class="reference-accessdate">. Retrieved <span class="nowrap">27 January</span> 2016</span>.</cite></span>

</li>

<li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><cite id="CITEREFWilcox,_Rand_R.2016" class="citation book cs1">Wilcox, Rand R. (2016). <i>Understanding and applying basic statistical methods using R</i>. Hoboken, New Jersey. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-119-06140-3</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/949759319">949759319</a>.</cite><span class="cs1-maint citation-comment"><code class="cs1-code">{{cite book}}</code>: CS1 maint: location missing publisher (link)</span></span>

</li>

<li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text">Singh, Laurie (2010) "Whither Efficient Markets? Efficient Market Theory and Behavioral Finance". The Finance Professionals' Post, 2010.</span>

</li>

<li id="cite_note-Edwards_2012_2-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-Edwards_2012_2_27-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFEdwards2012" class="citation journal cs1"><a href="Anthony_William_Fairbank_Edwards" class="mw-redirect" title="Anthony William Fairbank Edwards">Edwards, Anthony William Fairbank</a> (September 2012). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3430543">"Reginald Crundall Punnett: First Arthur Balfour Professor of Genetics, Cambridge, 1912"</a>. Perspectives. <i><a href="Genetics_(journal)" title="Genetics (journal)">Genetics</a></i>. <b>192</b> (1). Gonville and Caius College, Cambridge, UK: <a href="Genetics_Society_of_America" title="Genetics Society of America">Genetics Society of America</a>: <span class="nowrap">3–</span>13. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1534%2Fgenetics.112.143552">10.1534/genetics.112.143552</a>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3430543">3430543</a></span>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/22964834">22964834</a>. pp. <span class="nowrap">5–</span>6: <q>[…] Punnett's square seems to have been a development of 1905, too late for the first edition of his <i>Mendelism</i> (May 1905) but much in evidence in <i>Report III to the Evolution Committee of the Royal Society</i> [(Bateson et al. 1906b) "received March 16, 1906"]. The earliest mention is contained in a letter to Bateson from Francis Galton dated October 1, 1905 (Edwards 2012). We have the testimony of Bateson (1909, p. 57) that "For the introduction of this system [the 'graphic method'], which greatly simplifies difficult cases, I am indebted to Mr. Punnett." […] The first published diagrams appeared in 1906. […] when Punnett published the second edition of his <i>Mendelism</i>, he used a slightly different format ([…] Punnett 1907, p. 45) […] In the third edition (Punnett 1911, p. 34) he reverted to the arrangement […] with a description of the construction of what he called the "chessboard" method (although in truth it is more like a multiplication table). […]</q></cite> (11 pages)</span>

</li>

<li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><cite id="CITEREFGaoFongLiu2011" class="citation journal cs1">Gao, J.Z.; Fong, D.; Liu, X. (April 2011). "Mathematical analyses of casino rebate systems for VIP gambling". <i>International Gambling Studies</i>. <b>11</b> (1): <span class="nowrap">93–</span>106. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F14459795.2011.552575">10.1080/14459795.2011.552575</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:144540412">144540412</a>.</cite></span>

</li>

<li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><cite id="CITEREFGorman2010" class="citation journal cs1">Gorman, Michael F. (2010). "Management Insights". <i>Management Science</i>. <b>56</b>: <span class="nowrap">iv–</span>vii. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1287%2Fmnsc.1090.1132">10.1287/mnsc.1090.1132</a>.</cite></span>

</li>

<li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><cite id="CITEREFRoss2010" class="citation book cs1">Ross, Sheldon M. (2010). <i>A First course in Probability</i> (8th ed.). Pearson Prentice Hall. pp. <span class="nowrap">26–</span>27. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780136033134</bdi>.</cite></span>

</li>

<li id="cite_note-:2-31"><span class="mw-cite-backlink">^ <a href="#cite_ref-:2_31-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:2_31-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Probability.html">"Probability"</a>. <i>mathworld.wolfram.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">10 September</span> 2020</span>.</cite></span>

</li>

<li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text">Olofsson (2005) p. 8.</span>

</li>

<li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text">Olofsson (2005), p. 9</span>

</li>

<li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text">Olofsson (2005) p. 35.</span>

</li>

<li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text">Olofsson (2005) p. 29.</span>

</li>

<li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.statlect.com/fundamentals-of-probability/conditional-probability-as-a-random-variable">"Conditional probability with respect to a sigma-algebra"</a>. <i>statlect.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">4 July</span> 2022</span>.</cite></span>

</li>

<li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text">Riedi, P.C. (1976). Kinetic Theory of Gases-I. In: Thermal Physics. Palgrave, London. <a rel="nofollow" class="external free" href="https://doi.org/10.1007/978-1-349-15669-6_8">https://doi.org/10.1007/978-1-349-15669-6_8</a></span>

</li>

<li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text"><cite id="CITEREFBurgin2010" class="citation arxiv cs1">Burgin, Mark (2010). "Interpretations of Negative Probabilities". p. 1. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1008.1287v1">1008.1287v1</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/physics.data-an">physics.data-an</a>].</cite></span>

</li>

<li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><i>Jedenfalls bin ich überzeugt, daß der Alte nicht würfelt.</i> Letter to Max Born, 4 December 1926, in: <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LQIsAQAAIAAJ&q=achtung-gebietend">Einstein/Born Briefwechsel 1916–1955</a>.</span>

</li>

<li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text"><cite id="CITEREFMoore1992" class="citation book cs1">Moore, W.J. (1992). <i>Schrödinger: Life and Thought</i>. <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. p. 479. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-43767-7</bdi>.</cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2></div>

<ul><li><a href="Olav_Kallenberg" title="Olav Kallenberg">Kallenberg, O.</a> (2005) <i>Probabilistic Symmetries and Invariance Principles</i>. Springer-Verlag, New York. 510 pp. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-25115-4</bdi></li>

<li>Kallenberg, O. (2002) <i>Foundations of Modern Probability,</i> 2nd ed. Springer Series in Statistics. 650 pp. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-95313-2</bdi></li>

<li>Olofsson, Peter (2005) <i>Probability, Statistics, and Stochastic Processes</i>, Wiley-Interscience. 504 pp <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-67969-0</bdi>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

/* start https://en.wikipedia.org/ */

.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 .infobox .side-box{font-size:100%}.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}}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1237033735">

/* start https://en.wikipedia.org/ */

@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}}

/* end https://en.wikipedia.org/ */

</style><div class="side-box side-box-right sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">

/* start https://en.wikipedia.org/ */

.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}

/* end https://en.wikipedia.org/ */

</style>

<div class="side-box-text plainlist">Wikiquote has quotations related to <i><b><a href="https://en.wikiquote.org/wiki/Special:Search/Probability" class="extiw external" title="q:Special:Search/Probability">Probability</a></b></i>.</div></div>

</div>

<div class="side-box-text plainlist">Wikibooks has more on the topic of: <i><b><a href="https://en.wikibooks.org/wiki/Special:Search/Probability" class="extiw external" title="wikibooks:Special:Search/Probability">Probability</a></b></i></div></div>

</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:Probability" class="extiw external" title="commons:Category:Probability">Probability</a></span>.</div></div>

</div>

<div class="side-box-text plainlist">Wikisource has original works on the topic: <i><b><a href="https://en.wikisource.org/wiki/Portal:Probability_and_Statistics" class="extiw external" title="s:Portal:Probability and Statistics">Probability and Statistics</a></b></i></div></div>

</div>

<ul><li><a rel="nofollow" class="external text" href="http://www.math.uah.edu/stat/">Virtual Laboratories in Probability and Statistics (Univ. of Ala.-Huntsville)</a></li>

<li><a rel="nofollow" class="external text" href="https://www.bbc.co.uk/programmes/b00bqf61">Probability </a> on <a href="In_Our_Time_(radio_series)" title="In Our Time (radio series)"><i>In Our Time</i></a> at the <a href="BBC" title="BBC">BBC</a></li>

<li><a rel="nofollow" class="external text" href="http://wiki.stat.ucla.edu/socr/index.php/EBook">Probability and Statistics EBook</a></li>

<li><a href="Edwin_Thompson_Jaynes" title="Edwin Thompson Jaynes">Edwin Thompson Jaynes</a>. <i>Probability Theory: The Logic of Science</i>. Preprint: Washington University, (1996). – <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160119131820/http://omega.albany.edu:8008/JaynesBook.html">HTML index with links to PostScript files</a> and <a rel="nofollow" class="external text" href="http://bayes.wustl.edu/etj/prob/book.pdf">PDF</a> (first three chapters)</li>

<li><a rel="nofollow" class="external text" href="http://www.economics.soton.ac.uk/staff/aldrich/Figures.htm">People from the History of Probability and Statistics (Univ. of Southampton)</a></li>

<li><a rel="nofollow" class="external text" href="http://www.economics.soton.ac.uk/staff/aldrich/Probability%20Earliest%20Uses.htm">Probability and Statistics on the Earliest Uses Pages (Univ. of Southampton)</a></li>

<li><a rel="nofollow" class="external text" href="http://jeff560.tripod.com/stat.html">Earliest Uses of Symbols in Probability and Statistics</a> on <a rel="nofollow" class="external text" href="http://jeff560.tripod.com/mathsym.html">Earliest Uses of Various Mathematical Symbols</a></li>

<li><a rel="nofollow" class="external text" href="http://www.celiagreen.com/charlesmccreery/statistics/bayestutorial.pdf">A tutorial on probability and Bayes' theorem devised for first-year Oxford University students</a></li>

<li><a rel="nofollow" class="external text" href="http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html">Introduction to Probability – eBook</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110727200156/http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html">Archived</a> 27 July 2011 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, by Charles Grinstead, Laurie Snell <a rel="nofollow" class="external text" href="https://bitbucket.org/shabbychef/numas_text/">Source</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120325135243/https://bitbucket.org/shabbychef/numas_text/">Archived</a> 25 March 2012 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a> <i>(<a href="GNU_Free_Documentation_License" title="GNU Free Documentation License">GNU Free Documentation License</a>)</i></li>

<li><span class="languageicon">(in English and Italian)</span> <a href="Bruno_de_Finetti" title="Bruno de Finetti">Bruno de Finetti</a>, <i><a rel="nofollow" class="external text" href="http://amshistorica.unibo.it/35">Probabilità e induzione</a></i>, Bologna, CLUEB, 1993. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>88-8091-176-7</bdi> (digital version)</li>

<li><a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/I_06.html">Richard Feynman's Lecture on probability.</a></li></ul>

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}

/* end https://en.wikipedia.org/ */

</style></div><div role="navigation" class="navbox" aria-labelledby="Logic605" style="padding:3px"><table class="nowraplinks hlist mw-collapsible overview navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Logic605" style="font-size:114%;margin:0 4em"><a href="Logic" title="Logic">Logic</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><a href="History_of_logic" title="History of logic">History</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Major fields</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Logic_in_computer_science" title="Logic in computer science">Computer science</a></li>

<li><a href="Formal_semantics_(natural_language)" title="Formal semantics (natural language)">Formal semantics (natural language)</a></li>

<li><a href="Inference" title="Inference">Inference</a></li>

<li><a href="Philosophy_of_logic" title="Philosophy of logic">Philosophy of logic</a></li>

<li><a href="Formal_proof" title="Formal proof">Proof</a></li>

<li><a href="Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li>

<li><a href="Syntax_(logic)" title="Syntax (logic)">Syntax</a></li></ul>

</div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Logics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Classical_logic" title="Classical logic">Classical</a></li>

<li><a href="Informal_logic" title="Informal logic">Informal</a>

<ul><li><a href="Critical_thinking" title="Critical thinking">Critical thinking</a></li>

<li><a href="Reason" title="Reason">Reason</a></li></ul></li>

<li><a href="Mathematical_logic" title="Mathematical logic">Mathematical</a></li>

<li><a href="Non-classical_logic" title="Non-classical logic">Non-classical</a></li>

<li><a href="Philosophical_logic" title="Philosophical logic">Philosophical</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Argumentation_theory" title="Argumentation theory">Argumentation</a></li>

<li><a href="Metalogic" title="Metalogic">Metalogic</a></li>

<li><a href="Metamathematics" title="Metamathematics">Metamathematics</a></li>

</div></td></tr></tbody></table><div>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Foundations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Abductive_reasoning" title="Abductive reasoning">Abduction</a></li>

<li><a href="Analytic%E2%80%93synthetic_distinction" title="Analytic–synthetic distinction">Analytic and synthetic propositions</a></li>

<li><a href="Antecedent_(logic)" title="Antecedent (logic)">Antecedent</a></li>

<li><a href="Consequent" title="Consequent">Consequent</a></li>

<li><a href="Contradiction" title="Contradiction">Contradiction</a>

<ul><li><a href="Paradox" title="Paradox">Paradox</a></li>

<li><a href="Antinomy" title="Antinomy">Antinomy</a></li></ul></li>

<li><a href="Deductive_reasoning" title="Deductive reasoning">Deduction</a></li>

<li><a href="Deductive_closure" title="Deductive closure">Deductive closure</a></li>

<li><a href="Definition" title="Definition">Definition</a></li>

<li><a href="Description" title="Description">Description</a></li>

<li><a href="Dichotomy" title="Dichotomy">Dichotomy</a></li>

<li><a href="Logical_consequence" title="Logical consequence">Entailment</a>

<ul><li><a href="Linguistic_entailment" title="Linguistic entailment">Linguistic</a></li></ul></li>

<li><a href="Inductive_reasoning" title="Inductive reasoning">Induction</a></li>

<li><a href="Logical_truth" title="Logical truth">Logical truth</a></li>

<li><a href="Necessity_and_sufficiency" title="Necessity and sufficiency">Necessity and sufficiency</a></li>

<li><a href="Premise" title="Premise">Premise</a></li>

<li><a href="Proposition" title="Proposition">Proposition</a></li>

<li><a href="Reference" title="Reference">Reference</a></li>

<li><a href="Statement_(logic)" class="mw-redirect" title="Statement (logic)">Statement</a></li>

<li><a href="Substitution_(logic)" title="Substitution (logic)">Substitution</a></li>

<li><a href="Truth" title="Truth">Truth</a></li>

<li><a href="Validity_(logic)" title="Validity (logic)">Validity</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Lists</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Index_of_logic_articles" class="mw-redirect" title="Index of logic articles">Topics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="List_of_mathematical_logic_topics" title="List of mathematical logic topics">Mathematical logic</a></li>

<li><a href="List_of_Boolean_algebra_topics" title="List of Boolean algebra topics">Boolean algebra</a></li>

<li><a href="List_of_set_theory_topics" title="List of set theory topics">Set theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="List_of_logicians" title="List of logicians">Logicians</a></li>

<li><a href="List_of_rules_of_inference" title="List of rules of inference">Rules of inference</a></li>

<li><a href="List_of_paradoxes" title="List of paradoxes">Paradoxes</a></li>

<li><a href="List_of_fallacies" title="List of fallacies">Fallacies</a></li>

<li><a href="List_of_logic_symbols" title="List of logic symbols">Logic symbols</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> Category</li>

<li><span class="noviewer" typeof="mw:File"><span title="Outline"></span></span> <a href="Outline_of_logic" title="Outline of logic">Outline</a></li>

<li><span class="noviewer" typeof="mw:File"></span> <a href="Portal%3APhilosophy" title="Portal:Philosophy">Portal</a></li>

<li><span class="noviewer" typeof="mw:File"><span title="WikiProject"></span></span> WikiProject</li>

<li><a class="external text external" href="https://en.wikipedia.org/w/index.php?title=Special:Recentchangeslinked&target=Template:Logic&hidebots=0">changes</a></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Glossaries_of_science_and_engineering45" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Glossaries_of_science_and_engineering45" style="font-size:114%;margin:0 4em">Glossaries of <a href="Science" title="Science">science</a> and <a href="Engineering" title="Engineering">engineering</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Glossary_of_aerospace_engineering" title="Glossary of aerospace engineering">Aerospace engineering</a></li>

<li><a href="Glossary_of_agriculture" title="Glossary of agriculture">Agriculture</a></li>

<li><a href="Glossary_of_archaeology" title="Glossary of archaeology">Archaeology</a></li>

<li><a href="Glossary_of_architecture" title="Glossary of architecture">Architecture</a></li>

<li><a href="Glossary_of_artificial_intelligence" title="Glossary of artificial intelligence">Artificial intelligence</a></li>

<li><a href="Glossary_of_astronomy" title="Glossary of astronomy">Astronomy</a></li>

<li><a href="Glossary_of_biology" title="Glossary of biology">Biology</a></li>

<li><a href="Glossary_of_botanical_terms" title="Glossary of botanical terms">Botany</a></li>

<li><a href="Glossary_of_calculus" title="Glossary of calculus">Calculus</a></li>

<li><a href="Glossary_of_cell_biology" class="mw-redirect" title="Glossary of cell biology">Cell biology</a></li>

<li>Cellular and molecular biology

<li><a href="Glossary_of_chemistry_terms" title="Glossary of chemistry terms">Chemistry</a></li>

<li><a href="Glossary_of_civil_engineering" title="Glossary of civil engineering">Civil engineering</a></li>

<li><a href="Glossary_of_clinical_research" title="Glossary of clinical research">Clinical research</a></li>

<li><a href="Glossary_of_computer_hardware_terms" title="Glossary of computer hardware terms">Computer hardware</a></li>

<li><a href="Glossary_of_computer_science" title="Glossary of computer science">Computer science</a></li>

<li><a href="Glossary_of_developmental_biology" title="Glossary of developmental biology">Developmental and reproductive biology</a></li>

<li><a href="Glossary_of_ecology" title="Glossary of ecology">Ecology</a></li>

<li><a href="Glossary_of_economics" title="Glossary of economics">Economics</a></li>

<li><a href="Glossary_of_electrical_and_electronics_engineering" title="Glossary of electrical and electronics engineering">Electrical and electronics engineering</a></li>

<li>Engineering

<li><a href="Glossary_of_entomology_terms" title="Glossary of entomology terms">Entomology</a></li>

<li><a href="Glossary_of_environmental_science" title="Glossary of environmental science">Environmental science</a></li>

<li><a href="Glossary_of_genetics_and_evolutionary_biology" title="Glossary of genetics and evolutionary biology">Genetics and evolutionary biology</a></li>

<li>Geography

<li><a href="Glossary_of_Arabic_toponyms" title="Glossary of Arabic toponyms">Arabic toponyms</a></li>

<li><a href="Glossary_of_Hebrew_toponyms" title="Glossary of Hebrew toponyms">Hebrew toponyms</a></li>

<li><a href="Oikonyms_in_Western_and_South_Asia" title="Oikonyms in Western and South Asia">Western and South Asia</a></li></ul></li>

<li><a href="Glossary_of_geology" title="Glossary of geology">Geology</a></li>

<li><a href="Glossary_of_ichthyology" title="Glossary of ichthyology">Ichthyology</a></li>

<li><a href="Glossary_of_machine_vision" title="Glossary of machine vision">Machine vision</a></li>

<li><a href="Glossary_of_areas_of_mathematics" title="Glossary of areas of mathematics">Mathematics</a></li>

<li><a href="Glossary_of_mechanical_engineering" title="Glossary of mechanical engineering">Mechanical engineering</a></li>

<li><a href="Glossary_of_medicine" title="Glossary of medicine">Medicine</a></li>

<li><a href="Glossary_of_meteorology" title="Glossary of meteorology">Meteorology</a></li>

<li><a href="Glossary_of_mycology" title="Glossary of mycology">Mycology</a></li>

<li><a href="Glossary_of_nanotechnology" title="Glossary of nanotechnology">Nanotechnology</a></li>

<li><a href="Glossary_of_bird_terms" title="Glossary of bird terms">Ornithology</a></li>

<li><a href="Glossary_of_physics" title="Glossary of physics">Physics</a></li>

<li><a href="Glossary_of_probability_and_statistics" title="Glossary of probability and statistics">Probability and statistics</a></li>

<li><a href="Glossary_of_psychiatry" title="Glossary of psychiatry">Psychiatry</a></li>

<li><a href="Glossary_of_quantum_computing" title="Glossary of quantum computing">Quantum computing</a></li>

<li><a href="Glossary_of_robotics" title="Glossary of robotics">Robotics</a></li>

<li><a href="Glossary_of_scientific_naming" title="Glossary of scientific naming">Scientific naming</a></li>

<li><a href="Glossary_of_structural_engineering" title="Glossary of structural engineering">Structural engineering</a></li>

<li><a href="Glossary_of_virology" title="Glossary of virology">Virology</a></li></ul>

</div></td></tr></tbody></table></div>

/* start https://en.wikipedia.org/ */

.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}

/* end https://en.wikipedia.org/ */

</style>

/* start https://en.wikipedia.org/ */

.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}

/* end https://en.wikipedia.org/ */

</style></div><div role="navigation" class="navbox authority-control" aria-labelledby="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q9492#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata1299" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q9492#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata1299" style="font-size:114%;margin:0 4em">Authority control databases </div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">National</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Wahrscheinlichkeit"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4137007-7">Germany</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Probabilities"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85107090">United States</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="pravděpodobnost"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph360289&CON_LNG=ENG">Czech Republic</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007538702705171">Israel</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://lux.collections.yale.edu/view/concept/b4445cd7-29fd-48ee-b1ac-1297343429da">Yale LUX</a></span></li></ul></div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-06" href="https://en.wikipedia.org/wiki/?title=Probability&oldid=1299002974">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>