Statistics Dictionary

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Bayes Rule

Bayes' theorem (also known as Bayes' rule) is a 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 ), Baye's 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