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{{short description|Branch of mathematics concerning chance and uncertainty}}
{{Distinguish|probability theory|game theory|graph theory|statistics}}
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[[File:Dice Distribution (bar).svg|thumb|250px|The probabilities of rolling several numbers using two dice]]

'''Probability''' is a branch of [[mathematics]] and [[statistics]] concerning [[Event (probability theory)|events]] 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.{{NoteTag|Strictly speaking, a probability of 0 indicates that an event [[almost surely|''almost'' never]] takes place, whereas a probability of 1 indicates than an event [[almost surely|''almost'' certainly]] takes place. This is an important distinction when the [[sample space]] is infinite. For example, for the [[continuous uniform distribution]] on the [[real number|real]] 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 ''almost surely not'' be exactly 7. However, it does '''not''' mean that exactly 7 is ''impossible''. Ultimately some specific outcome (with probability 0) will be observed, and one possibility for that specific outcome is exactly 7.}}<ref name="Stuart and Ord 2009">"Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th ed., (2009), {{ISBN|978-0-534-24312-8}}.</ref><ref name="Feller">William Feller, ''An Introduction to Probability Theory and Its Applications'', vol.&nbsp;1, 3rd ed., (1968), Wiley, {{ISBN|0-471-25708-7}}.</ref> 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%).

These concepts have been given an [[Probability axioms|axiomatic]] mathematical formalization in ''[[probability theory]]'', which is used widely in [[areas of study]] such as [[statistics]], [[mathematics]], [[science]], [[finance]], [[gambling]], [[artificial intelligence]], [[machine learning]], [[computer science]], [[game theory]], and [[philosophy]] to, for example, draw inferences about the expected frequency of events. Probability theory is also used to describe the underlying mechanics and regularities of [[complex systems]].<ref>[http://www.britannica.com/EBchecked/topic/477530/probability-theory Probability Theory]. The Britannica website.</ref>

==Etymology==
{{see also|History of probability#Etymology|Glossary of probability and statistics}}
{{further|Likelihood}}

The word ''probability'' [[Etymology|derives]] from the Latin {{Lang|la|probabilitas}}, which can also mean "[[:wiktionary:probity|probity]]", a measure of the [[authority]] of a [[witness]] in a [[legal case]] in Europe, and often correlated with the witness's [[nobility]]. In a sense, this differs much from the modern meaning of ''probability'', which in contrast is a measure of the weight of [[empirical evidence]], and is arrived at from [[inductive reasoning]] and [[statistical inference]].<ref name="Emergence">[[Ian Hacking|Hacking, I.]] (2006) ''The Emergence of Probability: A Philosophical Study of Early Ideas about Probability, Induction and Statistical Inference'', Cambridge University Press, {{ISBN|978-0-521-68557-3}} {{page needed|date=June 2012}}</ref>

==Interpretations==
{{Main|Probability interpretations}}
When dealing with [[Experiment (probability theory)|random experiments]] – i.e., [[experiment]]s that are [[Randomness|random]] and [[Well-defined expression|well-defined]] – 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 '''theoretical probability''' (in contrast to [[empirical probability]], 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:
* [[Objectivity (philosophy)|Objectivists]] assign numbers to describe some objective or physical state of affairs. The most popular version of objective probability is [[frequentist probability]], which claims that the probability of a random event denotes the ''[[Frequency (statistics)|relative frequency]] of occurrence'' 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.<ref>{{cite book |title=The Logic of Statistical Inference |first=Ian |last=Hacking |author-link=Ian Hacking |year=1965 |publisher=Cambridge University Press |isbn=978-0-521-05165-1 }}{{page needed |date=June 2012 }}</ref> A modification of this is [[propensity probability]], which interprets probability as the tendency of some experiment to yield a certain outcome, even if it is performed only once.
* [[Subjective probability#Objective and subjective Bayesian probabilities|Subjectivists]] assign numbers per subjective probability, that is, as a [[Credence (statistics)|degree of belief]].<ref>{{cite journal |title=Logical foundations and measurement of subjective probability |first=Bruno de |last=Finetti |journal=Acta Psychologica |volume=34 |year=1970 |pages=129–145 |doi=10.1016/0001-6918(70)90012-0 }}</ref> 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",<ref>{{cite journal |last=Hájek |first=Alan|title=Interpretations of Probability |url = http://plato.stanford.edu/archives/win2012/entries/probability-interpret/ |journal = The Stanford Encyclopedia of Philosophy |edition = Winter 2012 |editor = Edward N. Zalta |access-date=22 April 2013 |date=2002-10-21 }}</ref> although that interpretation is not universally agreed upon.<ref>{{Cite book|section= Section A.2 The de Finetti system of probability |title=Probability Theory: The Logic of Science|last=Jaynes|first=E.T.|date=2003|publisher=Cambridge University Press|isbn=978-0-521-59271-0|editor-last=Bretthorst|editor-first=G. Larry|edition=1|language=en}}</ref> The most popular version of subjective probability is [[Bayesian probability]], which includes expert knowledge as well as experimental data to produce probabilities. The expert knowledge is represented by some (subjective) [[prior probability distribution]]. These data are incorporated in a [[likelihood function]]. The product of the prior and the likelihood, when normalized, results in a [[posterior probability distribution]] that incorporates all the information known to date.<ref>{{cite book |title=Introduction to Mathematical Statistics |first1=Robert V. |last1=Hogg |first2=Allen |last2=Craig |first3=Joseph W. |last3=McKean |edition=6th |year=2004 |location=Upper Saddle River |publisher=Pearson |isbn=978-0-13-008507-8 }}{{page needed|date=June 2012}}</ref> By [[Aumann's agreement theorem]], 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.<ref>{{Cite book|section= Section 5.3 Converging and diverging views |title=Probability Theory: The Logic of Science|last=Jaynes|first=E.T.|date=2003|publisher=Cambridge University Press|isbn=978-0-521-59271-0|editor-last=Bretthorst|editor-first=G. Larry|edition=1|language=en}}</ref>

== History ==
{{Main|History of probability}}

{{Further|History of statistics}}

The scientific study of probability is a modern development of mathematics. [[Gambling]] 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 {{NoteTag|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.}} are still obscured by superstitions.<ref>[[John E. Freund|Freund, John.]] (1973)  ''Introduction to Probability''. Dickenson {{isbn|978-0-8221-0078-2}} (p. 1)</ref>

According to [[Richard Jeffrey]], "Before the middle of the seventeenth century, the term 'probable' (Latin ''probabilis'') meant ''approvable'', 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."<ref name="Jeffrey">Jeffrey, R.C., ''Probability and the Art of Judgment,'' Cambridge University Press. (1992). pp. 54–55 . {{isbn|0-521-39459-7}}</ref> However, in legal contexts especially, 'probable' could also apply to propositions for which there was good evidence.<ref name="Franklin">Franklin, J. (2001) ''The Science of Conjecture: Evidence and Probability Before Pascal,'' Johns Hopkins University Press. (pp. 22, 113, 127)</ref>

[[File:Cardano.jpg|thumb|140px|[[Gerolamo Cardano]] (16th century)]]
[[File:Christiaan Huygens-painting.jpeg|thumb|140px|[[Christiaan Huygens]] published one of the first books on probability (17th century).]]

The sixteenth-century [[Italians|Italian]] polymath [[Gerolamo Cardano]] demonstrated the efficacy of defining [[odds]] 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<ref>{{cite web| url = http://www.columbia.edu/~pg2113/index_files/Gorroochurn-Some%20Laws.pdf| title = ''Some laws and problems in classical probability and how Cardano anticipated them'' Gorrochum, P. ''Chance'' magazine 2012}}</ref>).
Aside from the elementary work by Cardano, the doctrine of probabilities dates to the correspondence of [[Pierre de Fermat]] and [[Blaise Pascal]] (1654). [[Christiaan Huygens]] (1657) gave the earliest known scientific treatment of the subject.<ref>{{citation |url=http://www.secondmoment.org/articles/probability.php |publisher=Second Moment |access-date=2008-05-23 |title=A Brief History of Probability |last=Abrams |first=William |archive-date=24 July 2017 |archive-url=https://web.archive.org/web/20170724052656/http://www.secondmoment.org/articles/probability.php |url-status=dead }}</ref> [[Jakob Bernoulli]]'s ''[[Ars Conjectandi]]'' (posthumous, 1713) and [[Abraham de Moivre]]'s ''[[The Doctrine of Chances|Doctrine of Chances]]'' (1718) treated the subject as a branch of mathematics.<ref>{{Cite book  | last1 = Ivancevic | first1 = Vladimir G.
| last2 = Ivancevic | first2 = Tijana T.
| title = Quantum leap : from Dirac and Feynman, across the universe, to human body and mind
| year = 2008 | publisher = World Scientific
| location = Singapore; Hackensack, NJ | isbn = 978-981-281-927-7
| page = 16 }}</ref> See [[Ian Hacking]]'s ''The Emergence of Probability''<ref name=Emergence/> and [[James Franklin (philosopher)|James Franklin's]] ''The Science of Conjecture''<ref>{{cite book|last1=Franklin|first1=James|title=The Science of Conjecture: Evidence and Probability Before Pascal|date=2001|publisher=Johns Hopkins University Press|isbn=978-0-8018-6569-5}}</ref> for histories of the early development of the very concept of mathematical probability.

The [[theory of errors]] may be traced back to [[Roger Cotes]]'s ''Opera Miscellanea'' (posthumous, 1722), but a memoir prepared by [[Thomas Simpson]] in 1755 (printed 1756) first applied the theory to the discussion of errors of observation.<ref>{{Cite journal|last=Shoesmith|first=Eddie|date=November 1985|title=Thomas Simpson and the arithmetic mean|journal=Historia Mathematica|language=en|volume=12|issue=4|pages=352–355|doi=10.1016/0315-0860(85)90044-8|doi-access=free}}</ref> 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.

The first two laws of error that were proposed both originated with [[Pierre-Simon Laplace]]. 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{{snd}}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.<ref name=Wilson1923>Wilson EB (1923) "First and second laws of error". [[Journal of the American Statistical Association]], 18, 143</ref> 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."<ref name=Wilson1923 />

[[Daniel Bernoulli]] (1778) introduced the principle of the maximum product of the probabilities of a system of concurrent errors.
[[File:Bendixen - Carl Friedrich Gauß, 1828.jpg|thumb|140px|Carl Friedrich Gauss]]
[[Adrien-Marie Legendre]] (1805) developed the [[method of least squares]], and introduced it in his ''Nouvelles méthodes pour la détermination des orbites des comètes'' (''New Methods for Determining the Orbits of Comets'').<ref>{{cite web|last1=Seneta|first1=Eugene William|title="Adrien-Marie Legendre" (version 9)|url=http://statprob.com/encyclopedia/AdrienMarieLegendre.html|website=StatProb: The Encyclopedia Sponsored by Statistics and Probability Societies|access-date=27 January 2016|url-status=dead|archive-url=https://web.archive.org/web/20160203070724/http://statprob.com/encyclopedia/AdrienMarieLegendre.html|archive-date=3 February 2016}}</ref> In ignorance of Legendre's contribution, an Irish-American writer, [[Robert Adrain]], editor of "The Analyst" (1808), first deduced the law of facility of error,

<math display="block">\phi(x) = ce^{-h^2 x^2}</math>

where <math>h</math> is a constant depending on precision of observation, and <math>c</math> is a scale factor ensuring that the area under the curve equals 1. He gave two proofs, the second being essentially the same as [[John Herschel]]'s (1850).{{citation needed|date=June 2012}} [[Carl Friedrich Gauss|Gauss]] 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), [[James Ivory (mathematician)|James Ivory]] (1825, 1826), Hagen (1837), [[Friedrich Bessel]] (1838), [[William Fishburn Donkin|W.F. Donkin]] (1844, 1856), and [[Morgan Crofton]] (1870). Other contributors were [[Robert Leslie Ellis|Ellis]] (1844), [[Augustus De Morgan|De Morgan]] (1864), [[James Whitbread Lee Glaisher|Glaisher]] (1872), and [[Giovanni Schiaparelli]] (1875). [[Christian August Friedrich Peters|Peters]]'s (1856) formula{{clarify|date=June 2012}} for ''r'', the [[probable error]] of a single observation, is well known.

In the nineteenth century, authors on the general theory included [[Laplace]], [[Sylvestre Lacroix]] (1816), Littrow (1833), [[Adolphe Quetelet]] (1853), [[Richard Dedekind]] (1860), Helmert (1872), [[Hermann Laurent]] (1873), Liagre, Didion and [[Karl Pearson]]. [[Augustus De Morgan]] and [[George Boole]] improved the exposition of the theory.

In 1906, [[Andrey Markov]] introduced<ref>{{cite web|url = http://www.statslab.cam.ac.uk/~rrw1/markov/M.pdf|title = Markov Chains|first= Richard |last = Weber|website = Statistical Laboratory|publisher = University of Cambridge}}</ref> the notion of [[Markov chains]], which played an important role in [[stochastic process]]es theory and its applications. The modern theory of probability based on [[Measure (mathematics)|measure theory]] was developed by [[Andrey Kolmogorov]] in 1931.<ref>{{cite journal|last1=Vitanyi |first1= Paul M.B.|title=Andrei Nikolaevich Kolmogorov|journal=CWI Quarterly|date=1988|issue=1|pages=3–18|url=http://homepages.cwi.nl/~paulv/KOLMOGOROV.BIOGRAPHY.html|access-date=27 January 2016}}</ref>

On the geometric side, contributors to ''The Educational Times'' included Miller, Crofton, McColl, Wolstenholme, Watson, and [[Artemas Martin]].<ref>{{Cite book|last=Wilcox, Rand R.|title=Understanding and applying basic statistical methods using R|date= 2016|isbn=978-1-119-06140-3|location=Hoboken, New Jersey|oclc=949759319}}</ref> See [[integral geometry]] for more information.

==Theory==
{{Main|Probability theory}}
Like other [[theory|theories]], the [[probability theory|theory of probability]] is a representation of its concepts in formal terms{{snd}}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.

There have been at least two successful attempts to formalize probability, namely the [[Kolmogorov]] formulation and the [[Richard Threlkeld Cox|Cox]] formulation. In Kolmogorov's formulation (see also [[probability space]]), [[Set (mathematics)|sets]] are interpreted as [[Event (probability theory)|events]] and probability as a [[Measure (mathematics)|measure]] on a class of sets. In [[Cox's theorem]], 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 [[probability axioms|laws of probability]] are the same, except for technical details.

There are other methods for quantifying uncertainty, such as the [[Dempster–Shafer theory]] or [[possibility theory]], but those are essentially different and not compatible with the usually-understood laws of probability.

==Applications==
Probability theory is applied in everyday life in [[risk]] assessment and [[Statistical model|modeling]]. The insurance industry and [[Market (economics)|markets]] use [[actuarial science]] to determine pricing and make trading decisions. Governments apply probabilistic methods in [[environmental regulation]], entitlement analysis, and [[financial regulation]].

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 [[behavioral finance]] emerged to describe the effect of such [[groupthink]] on pricing, on policy, and on peace and conflict.<ref>Singh, Laurie (2010) "Whither Efficient Markets? Efficient Market Theory and Behavioral Finance". The Finance Professionals' Post, 2010.</ref>

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 [[Punnett squares]]).<ref name="Edwards_2012_2">{{cite journal |title=Reginald Crundall Punnett: First Arthur Balfour Professor of Genetics, Cambridge, 1912 |author-first=Anthony William Fairbank |author-last=Edwards |author-link=Anthony William Fairbank Edwards |location=Gonville and Caius College, Cambridge, UK |date=September 2012 |journal=[[Genetics (journal)|Genetics]] |department=Perspectives |publisher=[[Genetics Society of America]] |volume=192 |issue=1 |pages=3–13 |doi=10.1534/genetics.112.143552 |pmc=3430543 |pmid=22964834 |quote-pages=5–6 |quote=[…] Punnett's square seems to have been a development of 1905, too late for the first edition of his ''Mendelism'' (May 1905) but much in evidence in ''Report III to the Evolution Committee of the Royal Society'' [(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 ''Mendelism'', 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). […]}} (11 pages)</ref> 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 [[games of chance]] so that casinos can make a guaranteed profit, yet provide payouts to players that are frequent enough to encourage continued play.<ref>{{cite journal|last1=Gao|first1=J.Z.|last2=Fong|first2=D.|last3=Liu|first3=X.|title=Mathematical analyses of casino rebate systems for VIP gambling|journal=International Gambling Studies|date=April 2011|volume=11|issue=1|pages=93–106|doi=10.1080/14459795.2011.552575|s2cid=144540412}}<!--|access-date=13 October 2015--></ref>

Another significant application of probability theory in everyday life is [[reliability (statistics)|reliability]]. Many consumer products, such as [[automobiles]] 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 [[warranty]].<ref>{{Cite journal | doi=10.1287/mnsc.1090.1132| title=Management Insights| journal=Management Science| volume=56| pages=iv–vii| year=2010| last1=Gorman| first1=Michael F.| doi-access=}}</ref>

The [[cache language model]] and other [[Statistical Language Model|statistical language models]] that are used in [[natural language processing]] are also examples of applications of probability theory.

==Mathematical treatment==
[[File:probability vs odds.svg|thumb|right|Calculation of probability (risk) vs odds]]
{{see also|Probability axioms}}
Consider an experiment that can produce a number of results. The collection of all possible results is called the [[sample space]] of the experiment, sometimes denoted as <math>\Omega</math>. The [[power set]] 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 [[power set]] 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.

A probability is a [[Function (mathematics)|way of assigning]] 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.<ref>{{cite book|last = Ross|first =  Sheldon M. |title = A First course in Probability|edition= 8th |pages = 26–27|publisher = Pearson Prentice Hall|date = 2010|isbn = 9780136033134}}</ref>

The probability of an [[Event (probability theory)|event]] ''A'' is written as <math>P(A)</math>,<ref name=":2">{{Cite web|last=Weisstein|first=Eric W.|title=Probability|url=https://mathworld.wolfram.com/Probability.html|access-date=2020-09-10|website=mathworld.wolfram.com|language=en}}</ref> <math>p(A)</math>, or <math>\text{Pr}(A)</math>.<ref>Olofsson (2005) p. 8.</ref> This mathematical definition of probability can extend to infinite sample spaces, and even uncountable sample spaces, using the concept of a measure.

The ''opposite'' or ''complement'' of an event ''A'' is the event [not ''A''] (that is, the event of ''A'' not occurring), often denoted as <math>A', A^c</math>, <math>\overline{A}, A^\complement, \neg A</math>, or <math>{\sim}A</math>; its probability is given by {{nowrap|1= ''P''(not ''A'') = 1 − ''P''(''A'')}}.<ref>Olofsson (2005), p. 9</ref> As an example, the chance of not rolling a six on a six-sided die is {{nowrap|1=1 – (chance of rolling a six) =}} {{nowrap|1=1 − {{sfrac|1|6}} = {{sfrac|5|6}}.}} For a more comprehensive treatment, see [[Complementary event]].

If two events ''A'' and ''B'' occur on a single performance of an experiment, this is called the intersection or [[Joint distribution|joint probability]] of ''A'' and ''B'', denoted as <math>P(A \cap B).</math>

===Independent events===
[[File:Mutually Exclusive and Non-exclusive Probability Events.jpg|thumb|Events A and B depicted as mutually exclusive vs non-mutually exclusive in space Ω]]
If two events, ''A'' and ''B'' are [[Independence (probability theory)|independent]] then the joint probability is<ref name=":2" />
<math display="block" qid=Q120632573>P(A \mbox{ and }B) =  P(A \cap B) = P(A) P(B).</math>
For example, if two coins are flipped, then the chance of both being heads is <math>\tfrac{1}{2}\times\tfrac{1}{2} = \tfrac{1}{4}.</math><ref>Olofsson (2005) p. 35.</ref>

===Mutually exclusive events===
{{Main|Mutual exclusivity}}
If either event ''A'' or event ''B'' can occur but never both simultaneously, then they are called mutually exclusive events.

If two events are [[Mutually exclusive events|mutually exclusive]], then the probability of ''both'' occurring is denoted as <math>P(A \cap B)</math> and<math display="block">P(A \mbox{ and }B) =  P(A \cap B) = 0</math> If two events are [[Mutually exclusive events|mutually exclusive]], then the probability of ''either'' occurring is denoted as <math>P(A \cup B)</math> and<math display="block">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)</math>

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

===Not (necessarily) mutually exclusive events===

If the events are not (necessarily) mutually exclusive then<math display="block">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).</math> Rewritten,<math display="block">
P\left( A\cup B\right)  =P\left( A\right)  +P\left( B\right)  -P\left( A\cap B\right)  
</math>

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 <math>\tfrac{13}{52} + \tfrac{12}{52} - \tfrac{3}{52} = \tfrac{11}{26},</math> 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.

This can be expanded further for multiple not (necessarily) mutually exclusive events. For three events, this proceeds as follows:<math display="block"> 
\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} 
</math>It can be seen, then, that this pattern can be repeated for any number of events.

===Conditional probability===
''[[Conditional probability]]'' is the probability of some event ''A'', given the occurrence of some other event ''B''. Conditional probability is written <math>P(A \mid B)</math>, and is read "the probability of ''A'', given ''B''". It is defined by<ref>Olofsson (2005) p. 29.</ref>

<math display="block">P(A \mid B) = \frac{P(A \cap B)}{P(B)}\,</math>

If <math>P(B)=0</math> then <math>P(A \mid B)</math> is formally [[undefined (mathematics)|undefined]] by this expression. In this case <math>A</math> and <math>B</math> are independent, since <math>P(A \cap B) = P(A)P(B) = 0.</math> However, it is possible to define a conditional probability for some zero-probability events, for example by using a [[σ-algebra]] of such events (such as those arising from a [[continuous random variable]]).<ref>{{Cite web |title=Conditional probability with respect to a sigma-algebra |url=https://www.statlect.com/fundamentals-of-probability/conditional-probability-as-a-random-variable |access-date=2022-07-04 |website=statlect.com}}</ref>

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 <math>1/2;</math> 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 <math>1/3,</math> 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 <math>2/3.</math>

===Inverse probability===
{{Main|Inverse probability}}
In [[probability theory]] and applications, ''[[Bayes' theorem|Bayes' rule]]'' relates the [[odds]] of event <math>A_1</math> to event <math>A_2,</math> before (prior to) and after (posterior to) [[Conditional probability|conditioning]] on another event <math>B.</math> The odds on <math>A_1</math> to event <math>A_2</math> is simply the ratio of the probabilities of the two events. When arbitrarily many events <math>A</math> are of interest, not just two, the rule can be rephrased as ''posterior is proportional to prior times likelihood'', <math>P(A|B)\propto P(A) P(B|A)</math> where the proportionality symbol means that the left hand side is proportional to (i.e., equals a constant times) the right hand side as <math>A</math> varies, for fixed or given <math>B</math> (Lee, 2012; Bertsch McGrayne, 2012). In this form it goes back to Laplace (1774) and to Cournot (1843); see Fienberg (2005).

===Summary of probabilities===
{| class="wikitable plainrowheaders" style="text-align: left;"
|+Summary of probabilities
|-
! scope="col" | Event
! scope="col" | Probability
|-
! scope="row" style="text-align: center;" | A
| <math>P(A)\in[0,1]</math>
|-
! scope="row" style="text-align: center;" | not A
| <math>P(A^\complement)=1-P(A)\,</math>
|-
! scope="row" style="text-align: center;" | A or B
| <math>\begin{align}
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{align}</math>
|-
! scope="row" style="text-align: center;" | A and B
| <math>\begin{align}
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{align}</math>
|-
! scope="row" style="text-align: center;" | A given B
| <math>P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{P(B|A)P(A)}{P(B)} \,</math>
|}

==Relation to randomness and probability in quantum mechanics==
{{Main|Randomness}}
{{See also|Quantum fluctuation#Interpretations}}
In a [[determinism|deterministic]] universe, based on [[Newtonian mechanics|Newtonian]] concepts, there would be no probability if all conditions were known ([[Laplace's demon]]) (but there are situations in which [[chaos theory|sensitivity to initial conditions]] exceeds our ability to measure them, i.e. know them).  In the case of a [[roulette]] 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' [[eudaemons|Newtonian Casino]] 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 [[kinetic theory of gases]], where the system, while deterministic ''in principle'', is so complex (with the number of molecules typically the order of magnitude of the [[Avogadro constant]] {{val|6.02|e=23}}) that only a statistical description of its properties is feasible.<ref>Riedi, P.C. (1976). Kinetic Theory of Gases-I. In: Thermal Physics. Palgrave, London. https://doi.org/10.1007/978-1-349-15669-6_8</ref>

[[Probability theory]] is required to describe quantum phenomena.<ref>{{cite arXiv|last = Burgin|first= Mark|year =2010|title = Interpretations of Negative Probabilities|page= 1|class= physics.data-an|eprint=1008.1287v1}}</ref> A revolutionary discovery of early 20th century [[physics]] was the random character of all physical processes that occur at sub-atomic scales and are governed by the laws of [[quantum mechanics]]. The objective [[wave function]] evolves deterministically but, according to the [[Copenhagen interpretation]], it deals with probabilities of observing, the outcome being explained by a [[wave function collapse]] when an observation is made. However, the loss of [[determinism]] for the sake of [[instrumentalism]] did not meet with universal approval. [[Albert Einstein]] famously [[:de:Albert Einstein#Quellenangaben und Anmerkungen|remarked]] in a letter to [[Max Born]]: "I am convinced that God does not play dice".<ref>''Jedenfalls bin ich überzeugt, daß der Alte nicht würfelt.'' Letter to Max Born, 4 December 1926, in: [https://books.google.com/books?id=LQIsAQAAIAAJ&q=achtung-gebietend Einstein/Born Briefwechsel 1916–1955].</ref> Like Einstein, [[Erwin Schrödinger]], who [[Schrödinger equation#Historical background and development|discovered]] the wave function, believed quantum mechanics is a [[statistical]] approximation of an underlying deterministic [[reality]].<ref>{{cite book |last=Moore |first=W.J. |year=1992 |title=Schrödinger: Life and Thought |publisher=[[Cambridge University Press]] |page=479 |isbn= 978-0-521-43767-7}}</ref> In some modern interpretations of the statistical mechanics of measurement, [[quantum decoherence]] is invoked to account for the appearance of subjectively probabilistic experimental outcomes.

==See also==
{{Portal|Mathematics|Philosophy}}
{{main|Outline of probability}}
* [[Contingency (philosophy)|Contingency]]
* [[Equiprobability]]
* [[Fuzzy logic]]
* [[Heuristic (psychology)]]

== Notes ==
{{NoteFoot}}

== References ==
{{Reflist}}

==Bibliography==
* [[Olav Kallenberg|Kallenberg, O.]] (2005) ''Probabilistic Symmetries and Invariance Principles''. Springer-Verlag, New York. 510 pp.&nbsp;{{isbn|0-387-25115-4}}
* Kallenberg, O. (2002) ''Foundations of Modern Probability,'' 2nd ed. Springer Series in Statistics. 650 pp.&nbsp;{{isbn|0-387-95313-2}}
* Olofsson, Peter (2005) ''Probability, Statistics, and Stochastic Processes'', Wiley-Interscience. 504 pp {{isbn|0-471-67969-0}}.

==External links==
{{Wikiquote}}
{{Wikibooks}}
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{{wikisource portal|Probability and Statistics}}
{{Library resources box |by=no |onlinebooks=no |others=no |about=yes |label=Probability}}
* [http://www.math.uah.edu/stat/ Virtual Laboratories in Probability and Statistics (Univ. of Ala.-Huntsville)]
* {{Cite IEP|url-id=probability-and-induction|title=Probability and Induction}}
* {{In Our Time |Probability |b00bqf61 |Probability}}
* [http://wiki.stat.ucla.edu/socr/index.php/EBook Probability and Statistics EBook]
* [[Edwin Thompson Jaynes]]. ''Probability Theory: The Logic of Science''. Preprint: Washington University, (1996). – [https://web.archive.org/web/20160119131820/http://omega.albany.edu:8008/JaynesBook.html HTML index with links to PostScript files] and [http://bayes.wustl.edu/etj/prob/book.pdf PDF] (first three chapters)
* [http://www.economics.soton.ac.uk/staff/aldrich/Figures.htm People from the History of Probability and Statistics (Univ. of Southampton)]
* [http://www.economics.soton.ac.uk/staff/aldrich/Probability%20Earliest%20Uses.htm Probability and Statistics on the Earliest Uses Pages (Univ. of Southampton)]
* [http://jeff560.tripod.com/stat.html Earliest Uses of Symbols in Probability and Statistics] on [http://jeff560.tripod.com/mathsym.html Earliest Uses of Various Mathematical Symbols]
* [http://www.celiagreen.com/charlesmccreery/statistics/bayestutorial.pdf A tutorial on probability and Bayes' theorem devised for first-year Oxford University students]
* [http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html Introduction to Probability – eBook] {{Webarchive|url=https://web.archive.org/web/20110727200156/http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html |date=27 July 2011 }}, by Charles Grinstead, Laurie Snell [http://bitbucket.org/shabbychef/numas_text/ Source] {{Webarchive|url=https://web.archive.org/web/20120325135243/https://bitbucket.org/shabbychef/numas_text/ |date=25 March 2012 }} ''([[GNU Free Documentation License]])''
* {{in lang|en|it}} [[Bruno de Finetti]], ''[http://amshistorica.unibo.it/35 Probabilità e induzione]'', Bologna, CLUEB, 1993. {{isbn|88-8091-176-7}} (digital version)
* [https://feynmanlectures.caltech.edu/I_06.html Richard Feynman's Lecture on probability.]
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