Suppose A, B, and C are events of strictly positive probability in some probability space. If P(A | C) > P(B | C) and P(A | C) > P(B | C), is it true that P(A) > P(B)? If P(A | C) > P(A | C) and P(B | C) > P(B | C), is it true that P(A ∩ B | C) > P(A ∩ B | C)? [Hint]
2. Pólya’s urn scheme. In the urn model of Section 5, what is the probability that the first ball selected was black given that the second ball selected was black?