P(A | B) = P(A ^ B) / P(B)
P(B) = P(B | A)P(A)
P(A | B) = P(A ^ B) / P(B | A) / P(A)
In other words, even if A is a cause of B, we can consider them as correlated variables, and deduce the probability of A given B from knowing the probabilities of B given A, the probability of A (the cause) all by itself, and the probability of A and B (when the cause results in the effect).
There has been some argument (see Wikipedia), however, about assigning the "a priori" probabilities P(B | A). Is that valid, considering we are measuring P (A | B ) ?