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Conditional Probability

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The conditional probability of an event A assuming that B has occurred, denoted P(A|B), equals

P(A|B)=(P(A intersection B))/(P(B)),
(1)

which can be proven directly using a Venn diagram. Multiplying through, this becomes

P(A|B)P(B)=P(A intersection B),
(2)

which can be generalized to

P(A intersection B intersection C)=P(A)P(B|A)P(C|A intersection B).
(3)

Rearranging (1) gives

P(B|A)=(P(B intersection A))/(P(A)).
(4)

Solving (4) for P(B intersection A)=P(A intersection B) and plugging in to (1) gives

P(A|B)=(P(A)P(B|A))/(P(B)).
(5)

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