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If two events A and B are independent, then the conditional probability of A given B is the same as the "unconditional" (or "marginal") probability of A, i.e.,

There are at léast two réasons why this statement is not taken to be the definition of independence: (1) the two events A and B do not play symmetrical roles in this statement, and (2) problems arise with this statement when events of probability 0 are involved.

When one recalls that the conditional probability P(A | B) is given by

one sees that the statement above is equivalent to

Here A ∩ B is the intersection of A and B, i.e., it is the event that both events A and B occur. Thus we could say:

Thus the standard definition says:

Two events A and B are independent iff P(A ∩ B)=P(A)P(B).

More generally, and collection of events—possibly more than just two of them—are mutually independent precisely if for any finite subset A1, ..., An of the collection we have

This is called the multiplication rule for independent events.

If any two of a collection of random variables are independent, they may nonetheless fail to be mutually independent; this is called pairwise independence.