Suppose we are interested in the probability that some event A occurs, and we learn that the event B occurred. How should we update the probability of A to reflect this new knowledge? This is what the conditional probability does: it says how the additional knowledge that B occurred should affect the probability that A occurred quantitatively. For example, suppose that A and B are mutually exclusive. Then if B occurred, A did not, so the conditional probability that A occurred given that B occurred is zero. At the other extreme, suppose that B is a subset of A, so that A must occur whenever B does. Then if we learn that B occurred, A must have occurred too, so the conditional probability that A occurred given that B occurred is 100%. For in-between cases, where A and B intersect, but B is not a subset of A, the conditional probability of A given B is a number between zero and 100%. Basically, one "restricts" the outcome space S to consider only the part of S that is in B, because we know that B occurred. For A to have happened given that B happened requires that AB happened, so we are interested in the event AB. To have a legitimate probability requires that P(S) = 100%, so if we are restricting the outcome space to B, we need to divide by the probability of B to make the probability of this new S be 100%. On this scale, the probability that AB happened is P(AB)/P(B). This is the definition of the conditional probability of A given B, provided P(B) is not zero (division by zero is undefined). Note that the special cases AB = {} (A and B are mutually exclusive) and AB = B (B is a subset of A) agree with our intuition as described at the top of this paragraph. Conditional probabilities satisfy the axioms of probability, just as ordinary probabilities do.