Conditional probability: Difference between revisions

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==== [[Andrey Kolmogorov|Kolmogorov]] definition ====

Given two [[event (probability theory)|events]] {{mvar|A}} and {{mvar|B}} from the [[sigma-field]] of a probability space, with the [[marginal probability|unconditional probability]] of {{mvar|B}} being greater than zero (i.e., {{math|P(''B'') > 0)}}, the conditional probability of {{mvar|A}} given {{mvar|B}} (<math>P(A \mid B)</math>) is the probability of ''A'' occurring if ''B'' has or is assumed to have happened.<ref name=":1">{{Cite book|last=Reichl|first=Linda Elizabeth|title=A Modern Course in Statistical Physics|publisher=WILEY-VCH|year=2016|isbn=978-3-527-69049-7|edition=4th revised and updated|chapter=2.3 Probability}}</ref> ''A'' is assumed to be athe set of all possible outcomes of an experiment or random trial that has a restricted or reduced sample space. The conditional probability can be found by the [[quotient]] of the probability of the joint intersection of events {{mvar|A}} and {{mvar|B}} (<math>P(A \cap B)</math>)—the probability at which ''A'' and ''B'' occur together, although not necessarily occurring at the same time—and the [[probability]] of {{mvar|B}}:<ref name=":0" /><ref>{{citation|last=Kolmogorov|first=Andrey|title=Foundations of the Theory of Probability|publisher=Chelsea|year=1956 }}</ref><ref>{{Cite web|title=Conditional Probability|url=http://www.stat.yale.edu/Courses/1997-98/101/condprob.htm|access-date=2020-09-11|website=www.stat.yale.edu}}</ref>

:<math>P(A \mid B) = \frac{P(A \cap B)}{P(B)}</math>.