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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|
It can be shown that
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is also equivalent. Although the derived forms may seem more intuitive, they are not the preferred definition as the conditional probabilities may be undefined, and the preferred definition is symmetrical in ''A'' and ''B''. Independence does not refer to a disjoint event.<ref>{{Cite book|last=Tijms|first=Henk|url=https://www.cambridge.org/core/books/understanding-probability/B82E701FAAD2C0C2CF36E05CFC0FF3F2|title=Understanding Probability|date=2012|publisher=Cambridge University Press|isbn=978-1-107-65856-1|edition=3|___location=Cambridge|doi=10.1017/cbo9781139206990}}</ref>
It should also be noted that given the independent event pair [''A'',''B''] and an event ''C'', the pair is defined to be [[Conditional independence|conditionally independent]] if<ref>{{Cite book|last=Pfeiffer|first=Paul E.
: <math>P(AB \mid C) = P(A \mid C)P(B \mid C).</math>
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