Let A and B be two events in a sample space . Prove that P(A|B) > P(A) holds if and only if P(B|A) > P(B).

Let A and B be two events in a sample space Ω. Prove that P(A|B) > P(A) holds if and only if P(B|A) > P(B). In such a case, the two events A and B are said to be positively correlated since the knowledge that one has appeared increases the probability that the other appears, too.

Verify also the dual statement: P(A|B) < P(A) holds if and only if P(B|A) < P(B)
(in which case, we say that A and B are negatively correlated).
Application: When selecting a card twice in succession, we define the events A: at least one ace turns up;
B: the first two outcomes are different;
C: the first outcome is not an ace.
Examine whether each of the two pairs of events (A, B) and (A,C) are either positively or negatively correlated.