QuestionThe probability, P(A ∩ B), that both A and B happen seem closely related to P(A|B), that A happens given that B happens. Are they related? If so how?
AnswerP(A ∩ B) and P(A|B) are very closely related. Their only difference is that the conditional probability assumes that we already know something -- that B is true. The intersection doesn't assume that we know anything. So for P(A ∩ B), we will receive a probability between 0, impossible, and 1, certain. For P(A|B), however, we will receive a probability between 0, if A cannot happen when B is true, and P(B), if A is always true when B is true.
So the only difference between P(A ∩ B) and P(A|B) is the number P(B).