The belief in an event given the certainty of another event. A conditional probability is written P(A|B), where A is the unknown event and B is the known event. The expression P(A|B) is sometimes spoke as "the probability of A conditioned on B".

The traditional definition of conditional probabilities is:

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

where P(A,B) is the probabilty that A and B both occur. You can see this by thinking of the

combinations of all outcomes of A and B:

B ¬ B
-----------------------------
A | P(A,B) | P(A,¬B) | = P(A)
-----------------------------
¬A | P(¬A,B) | P(¬A,¬B) | = P(¬A)
-----------------------------
= P(A) = P(¬B)

The sums of the rows and columns (i.e., the marginal probabilities) remove the effects of one of the variables.

So, what's the value of P(A|B)? By definition, this means that we know for sure that B has occurred, so we can consider only the first column of the table (B) and ignore the second column (¬B). At this point, this is the space of all possible events, so it must sum to 1, ¬ P(A), hence we divide each item in the column by P(A). We're now interested in the probability that A occurs, so we take the value of the first box, P(A,B), which coincidentally has been divided by P(B).

There's a more philosophical approach to conditional probabilities, however. One can consider the conditioning variable, B, as someone's background knowledge or frame of reference, and the conditional, A|B, as an event A in the context of this background knowledge. Bayes' Theorem, which makes use of these terms, is viewed as a fundamental method for updating beliefs given evidence.