In this problem, we will say that event A attracts event B if P(B | A) > P(B), and event A repels event B if P(B | A) < P(B). For simplicity, you may assume that the probability of all events is greater than zero.

a) Show that if A attracts B, then B must also attract A.

b) Show that if A neither attracts nor repels B, A and B must be independent.

c) Show that if A attracts B, then A repels B^c .

d) Show that if A attracts B, then P(B | A) > P(B | A^c ).