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.

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