Generalizations

Bayes theorem for 3 events

A version of Bayes' theorem for 3 events results from the addition of a third event \(C\), with \(P(C)>0\), on which all probabilities are conditioned:

\(P(A\vert B\cap C)={\frac {P(B\vert A\cap C)\,P(A\vert C)}{P(B\vert C)}}\)

Derivation

Using the chain rule

\(P(A\cap B\cap C)=P(A\vert B\cap C)\,P(B\vert C)\,P(C)\)

And, on the other hand

\(P(A\cap B\cap C)=P(B\cap A\cap C)=P(B\vert A\cap C)\,P(A\vert C)\,P(C)\)

The desired result is obtained by identifying both expressions and solving for \(P(A\vert B\cap C)\).