In probability theory, The Multiplication Rule states that the probability that A and B occur is equal to the probability that A occurs times the probability that B occurs, given that we know A has already occurred. This rule can be written:

P(A∩B) = P(B) x P(A|B),

or alternatively as:

P(A∩B) = P(A) x P(B|A).

As illustrated in , the events A and B are defined on a sample space, which is the set of all possible outcomes of an experiment.

We obtain the general multiplication rule by multiplying both sides of the definition of conditional probability by the denominator. In general, the conditional probability can be defined as follows:

If P(B) ≠ 0 then the conditional probability of A relative to B is given by

P(A|B) = P(A∩B) x P(B).

If P(A|B) = P(A) then we say that A and B are independent events.

As an example, suppose that we draw two cards out of a deck of cards and let A = {first card is an ace}, and B = {second card is an ace}, then

and

The denominator in the latter equation is 51 since we know a card has been drawn already. Therefore, there are 51 left in total. We also know the first card was an ace, therefore:

P(A∩B) = P(A) x P(B|A),

P(A∩B) =