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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 the 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}}, that is, <math>P(A \cap B)</math>, the probability at which ''A'' and ''B'' occur together, 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>
For a sample space consisting of equal likelihood outcomes, the probability of the event ''A'' is understood as the fraction of the number of outcomes in ''A'' to the number of all outcomes in the sample space. Then, this equation is understood as the fraction of the set <math>A \cap B</math> to the set ''B''. Note that the above equation is a definition, not just a theoretical result. We denote the quantity <math>\frac{P(A \cap B)}{P(B)}</math> as <math>P(A\mid B)</math> and call it the "conditional probability of {{mvar|A}} given {{mvar|B}}."
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Some authors, such as [[Bruno de Finetti|de Finetti]], prefer to introduce conditional probability as an [[Probability axioms|axiom of probability]]:
:<math>P(A \cap B) = P(A \mid B)P(B). </math>
This equation for a conditional probability, although mathematically equivalent, may be intuitively easier to understand. It can be interpreted as "the probability of ''B'' occurring multiplied by the probability of ''A'' occurring, provided that ''B'' has occurred, is equal to the probability of the ''A'' and ''B'' occurrences together, although not necessarily occurring at the same time". Additionally, this may be preferred philosophically; under major [[probability interpretations]], such as the [[Subjective probability|subjective theory]], conditional probability is considered a primitive entity. Moreover, this "multiplication rule" can be practically useful in computing the probability of <math>A \cap B</math> and introduces a symmetry with the summation axiom for Poincaré Formula:
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:<math>P(A \cup B) = P(A) + P(B) - P(A \cap B)</math>
:Thus the equations can be combined to find a new representation of the :
:<math> P(A \cap B)= P(A) + P(B) - P(A \cup B) = P(A \mid B)P(B) </math>
:<math> P(A \cup B)= {P(A) + P(B) - P(A \mid B){P(B)}} </math>▼
▲:<math> P(A \cup B)= {P(A) + P(B) - P(A \mid B){P(B)}}
==== As the probability of a conditional event ====
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Conditional probability can be defined as the probability of a conditional event <math>A_B</math>. The [[Goodman–Nguyen–Van Fraassen algebra|Goodman–Nguyen–Van Fraassen]] conditional event can be defined as:
:<math>A_B = \bigcup_{i \ge 1} \left( \bigcap_{j<i} \overline{B}_j, A_i B_i \right), </math> where <math>A_i </math> and <math>B_i </math> represent states or elements of ''A'' or ''B.'' <ref>{{Cite journal|last1=Flaminio|first1=Tommaso|last2=Godo|first2=Lluis|last3=Hosni|first3=Hykel|date=2020-09-01|title=Boolean algebras of conditionals, probability and logic|url=https://www.sciencedirect.com/science/article/pii/S000437022030103X|journal=Artificial Intelligence|language=en|volume=286|pages=103347|doi=10.1016/j.artint.2020.103347|arxiv=2006.04673|s2cid=214584872 |issn=0004-3702}}</ref>▼
▲</math> represent states or elements of ''A'' or ''B.'' <ref>{{Cite journal|last1=Flaminio|first1=Tommaso|last2=Godo|first2=Lluis|last3=Hosni|first3=Hykel|date=2020-09-01|title=Boolean algebras of conditionals, probability and logic|url=https://www.sciencedirect.com/science/article/pii/S000437022030103X|journal=Artificial Intelligence|language=en|volume=286|pages=103347|doi=10.1016/j.artint.2020.103347|arxiv=2006.04673|s2cid=214584872 |issn=0004-3702}}</ref>
It can be shown that
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:<math>P(A_B)= \frac{P(A \cap B)}{P(B)}</math>
which meets the Kolmogorov definition of conditional probability.<ref>{{Citation|last=Van Fraassen|first=Bas C.|title=Probabilities of Conditionals|date=1976|url=https://doi.org/10.1007/978-94-010-1853-1_10|work=Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science: Volume I Foundations and Philosophy of Epistemic Applications of Probability Theory|pages=261–308|editor-last=Harper|editor-first=William L.|series=The University of Western Ontario Series in Philosophy of Science|place=Dordrecht|publisher=Springer Netherlands|language=en|doi=10.1007/978-94-010-1853-1_10|isbn=978-94-010-1853-1|access-date=2021-12-04|editor2-last=Hooker|editor2-first=Clifford Alan|url-access=subscription}}</ref>
=== Conditioning on an event of probability zero ===
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