Statistics and Probability Dictionary

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Bayes Rule

Bayes' theorem (also known as Bayes' rule) is a very useful tool for calculating conditional probabilities . Bayes' theorem can be stated as follows:

Bayes' theorem. Let A1, A2, ... , An be a set of mutually exclusive events that together form the sample space S. Let B be any event from the same sample space, such that P(B) > 0. Then,

P( Ak | B ) = P( Ak ∩ B )
P( A1 ∩ B ) + P( A2 ∩ B ) + . . . + P( An ∩ B )

Note: Invoking the fact that P( Ak ∩ B ) = P( Ak )P( B | Ak ), Bayes' theorem can also be expressed as

P( Ak | B ) = P( Ak ) P( B | Ak )
P( A1 ) P( B | A1 ) + P( A2 ) P( B | A2 ) + . . . + P( An ) P( B | An )


See also:   Statistics Tutorial: Bayes' Theorem | Rules of Probability | Bayes Rule Calculator