Bayes' Theorem

Notation:
Events are designated by capital letters
P(A) represents the probability that event A occurs
P(A,B) represents the probability that both A and B occur
P(A|B) represents the probability that A occurs given that B occurs

P(A,B) = P(A)xP(B|A) [intuitively obvious]
also P(A,B) = P(B)xP(A|B)
so P(A)xP(B|A) = P(B)xP(A|B) because both are equal to P(A,B)
divide both sides by P(A) giving

P(B|A) = P(B)xP(A|B)/P(A) qed

actually often stated with A and B reversed

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

P(A) is often determined by the sum of P(Bi)xP(A|Bi) over all i.

For continuous functions the sum is replaced by an integral.