Bayes' Theorem can also be used as the following :

$P(A | B) = \frac{P(A \cap B)}{P(B)}$ The way to think about this is that given two events A and B, the probability that A happens, given B occurs, is the probability of their intersection occuring i.e. both events occuring, over the probability that B occurs. The easiest way to see why this works is for Independent events. If A and B are independent, the probability that they both occur is the product of their individual probabilities. Which then simplifies the above equation to : $P(A|B) = \frac{P(A \cap B)}{P(B)}$ $\implies P(A|B) = \frac{P(A).P(B)}{P(B)}$ $\implies P(A|B) = P(A)$ Which makes sense since the events are independent.