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{{about||the mathematical field of probability specifically rather than a general discussion|Probability theory|other uses}}
{{Distinguish|Probably (disambiguation){{!}}Probably}}

{{use dmy dates|date=October 2013}}

{{ProbabilityTopics}}
{{Certainty}}
[[File:Dice Probability.png|thumb|250px|The probabilities of rolling several numbers using two dice.]]
'''Probability''' is a [[measure (mathematics)|measure]] quantifying the [[likelihood]] that [[Event (probability theory)|events]] will occur.<ref>[https://web.archive.org/web/20150428142545/http://machaut.uchicago.edu/?resource=Webster%27s&word=probability&use1913=on "Probability"]. ''[[Webster's Dictionary|Webster's Revised Unabridged Dictionary]]''. G & C Merriam, 1913.</ref> Probability is a number between 0 and 1, where, roughly  speaking,{{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 when we make an observation, it 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.}} 0 indicates impossibility and 1 indicates certainty.<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 1),  3rd Ed, (1968), Wiley, {{ISBN|0-471-25708-7}}.</ref> The higher the probability of an event, the more likely it is that the event will occur. 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 such [[areas of study]] as [[mathematics]], [[statistics]], [[finance]], [[gambling]], [[science]] (in particular [[physics]]), [[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>

==Interpretations==
{{Main|Probability interpretations}}
When dealing with [[experiment]]s that are [[random]] and [[well-defined]] in a purely theoretical setting (like tossing a fair coin), probabilities can be numerically described by the number of desired outcomes divided by the total number of all outcomes. For example, tossing a fair 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%. However, when it comes to practical application, there are two major competing categories of probability interpretations, whose adherents possess 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 ''relative frequency of occurrence'' of an experiment's outcome, when repeating the experiment. 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 |authorlink=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, i.e., as a degree of belief.<ref>{{cite journal |title=Logical foundations and measurement of subjective probability |first=Bruno de |last=Finetti |journal=Acta Psychologica |volume=34 |issue= |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 |accessdate=22 April 2013 |date=2002-10-21 }}</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, normalized, results in a [[posterior probability distribution]] that incorporates all the information known to date.<ref>{{cite book |title=Introduction to Mathematical Statistics |first=Robert V. |last=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=English}}</ref>

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

The word ''probability'' [[Etymology|derives]] from the Latin ''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>

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

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 for millennia, 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, {{Clarify|text=fundamental issues|date=July 2014}} are still obscured by the superstitions of gamblers.<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, unequivocally, 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>

[[Image:Al-kindi cryptographic.png|thumb|140px|[[Al-Kindi]]'s ''Book of Cryptographic Messages'' contains the earliest known use of [[statistical inference]] (9th century)]]

The earliest known forms of [[probability and statistics]] were developed by [[Mathematics in medieval Islam|Middle Eastern mathematicians]] studying [[cryptography]] between the 8th and 13th centuries. [[Al-Khalil ibn Ahmad al-Farahidi|Al-Khalil]] (717–786) wrote the ''Book of Cryptographic Messages'' which contains the first use of [[permutations and combinations]] to list all possible [[Arabic language|Arabic]] words with and without vowels. [[Al-Kindi]] (801–873) made the earliest known use of [[statistical inference]] in his work on [[cryptanalysis]] and [[frequency analysis]]. An important contribution of [[Ibn Adlan]] (1187–1268) was on [[sample size]] for use of frequency analysis.<ref name="LB">{{cite journal|last=Broemeling|first=Lyle D.|title=An Account of Early Statistical Inference in Arab Cryptology|journal=The American Statistician|date=1 November 2011|volume=65|issue=4|pages=255–257|doi=10.1198/tas.2011.10191}}</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>[http://www.columbia.edu/~pg2113/index_files/Gorroochurn-Some%20Laws.pdf ''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 |accessdate=2008-05-23 |title=A Brief History of Probability |last=Abrams |first=William }}</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}}</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, 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|accessdate=27 January 2016|url-status=dead|archiveurl=https://web.archive.org/web/20160203070724/http://statprob.com/encyclopedia/AdrienMarieLegendre.html|archivedate=3 February 2016|df=dmy-all}}</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>\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), [[W.F. Donkin]] (1844, 1856), and [[Morgan Crofton]] (1870). Other contributors were 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.

[[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]] (1906), which played an important role in [[stochastic process]]es theory and its applications. The modern theory of probability based on the [[Measure (mathematics)|measure theory]] was developed by [[Andrey Kolmogorov]] (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|accessdate=27 January 2016}}</ref>

On the geometric side (see [[integral geometry]]) contributors to ''The Educational Times'' were influential (Miller, Crofton, McColl, Wolstenholme, Watson, and [[Artemas Martin]]).{{Citation needed|date=February 2012}}

{{Further|History of statistics}}

==Theory==
{{Main|Probability theory}}
Like other [[theory|theories]], the [[probability theory|theory of probability]] 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.

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 [[probability space]]), [[Set (mathematics)|sets]] are interpreted as [[Event (probability theory)|events]] and probability itself as a [[Measure (mathematics)|measure]] on a class of sets. In [[Cox's theorem]], probability is taken as a primitive (that is, 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 laws of probability as usually understood.

==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 ([[Reliability theory of aging and longevity]]), and [[financial regulation]].

A good 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 very 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). 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}}<!--|accessdate=13 October 2015--></ref>

The discovery of rigorous methods to assess and combine probability assessments has changed society.<ref name=":0">{{Cite web|url=http://archon.educ.kent.edu/Oasis/Resc/Educ/data.html|title=Data: Data Analysis, Probability and Statistics, and Graphing|website=archon.educ.kent.edu|access-date=2017-05-28}}</ref>{{Citation needed|date=October 2015}}

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.}}</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==
{{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. The [[power set]] of the sample space is formed by considering all different collections of possible results. For example, rolling a dice can produce six possible results. One collection of possible results gives an odd number on the dice. 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 dice 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 if you look at a collection of mutually exclusive events (events with no common results, e.g., the events {1,6}, {3}, and {2,4} are all mutually exclusive), 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>, <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>\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 – (chance of rolling a six) }}<math>= 1 - \tfrac{1}{6} = \tfrac{5}{6}</math>. See [[Complementary event]] for a more complete treatment.

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===

If two events, ''A'' and ''B'' are [[Independence (probability theory)|independent]] then the joint probability is
:<math>P(A \mbox{ and }B) =  P(A \cap B) = P(A) P(B),\,</math>
for example, if two coins are flipped 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===

If either event ''A'' or event ''B'' but never both occurs on a single performance of an experiment, 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>.
:<math>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>.
:<math>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 {{dice}} is <math>P(1\mbox{ or }2) = P(1) + P(2) = \tfrac{1}{6} + \tfrac{1}{6} = \tfrac{1}{3}.</math>

===Not mutually exclusive events===

If the events are not mutually exclusive then
:<math>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>
For example, when drawing a single card at random from a regular deck of cards, the chance of getting a heart or a face card (J,Q,K) (or one that is both) is <math>\tfrac{13}{52} + \tfrac{12}{52} - \tfrac{3}{52} = \tfrac{11}{26}</math>, because of 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.

===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>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. However, it is possible to define a conditional probability for some zero-probability events using a [[σ-algebra]] of such events (such as those arising from a [[continuous random variable]]).{{citation needed|date=July 2012}}

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, such as, if a red ball was taken, 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.

===Inverse probability===
In [[probability theory]] and applications, '''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). See [[Inverse probability]] and [[Bayes' rule]].

===Summary of probabilities===
{| class="wikitable"
|+Summary of probabilities
|-
!Event!!Probability
|-
|align=center|A||<math>P(A)\in[0,1]\,</math>
|-
|align=center|not A||<math>P(A^\complement)=1-P(A)\,</math>
|-
|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>
|-
|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>
|-
|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 [[kinetic theory of gases|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.

[[Probability theory]] is required to describe quantum phenomena.<ref>{{cite journal|last = Burgin|first= Mark|year =2010|title = Interpretations of Negative Probabilities|page= 1|arxiv=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}}
* [[Chance (disambiguation)]]
* [[Class membership probabilities]]
* [[Contingency (philosophy)|Contingency]]
* [[Equiprobability]]
* [[Heuristics in judgment and decision-making]]
* [[Probability theory]]
* [[Randomness]]
* [[Statistics]]
* [[Estimators]]
* [[Estimation Theory]]
* [[Probability density function]]
;In [[Law]]
* [[Balance of probabilities]]

== 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}}
{{Commons category}}
{{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)]
* {{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://ubu.com/historical/young/index.html] pdf file of [[An Anthology of Chance Operations]] (1963) at [[UbuWeb]]
* [http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html Introduction to Probability – eBook], by Charles Grinstead, Laurie Snell [http://bitbucket.org/shabbychef/numas_text/ Source] ''([[GNU Free Documentation License]])''
* {{en icon}} {{it icon}} [[Bruno de Finetti]], ''[http://amshistorica.unibo.it/35 Probabilità e induzione]'', Bologna, CLUEB, 1993. {{isbn|88-8091-176-7}} (digital version)
* [http://www.feynmanlectures.caltech.edu/I_06.html Richard P. Feynman's Lecture on probability.]
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