Bayes' Theorem

Let P be a function giving probabilities of events. If P(A | B) denotes the probability of event A assuming event B:

                P(A and B) = P(A | B) P(B) = P(B | A) P(A).

Therefore, 

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

This is referred to as Bayes' Theorem. For example, assume A is the event of getting a six and B of getting an even when rolling a die:

                P(A) = 1 / 6

                P(B) = 1 / 2

                P(A | B) = 1 / 3

                P(B | A) = 1

                P(A and B) = 1 / 6.

Note P(A | B) does not equal P(A)!