1. Prove each of the following statements. (Assume that any conditioning event has positive probability)

(a) If P(B) = 1, then P(A|B) = P(A) for any A.

(b) If A B, then P(B|A) = 1 and P(A|B) = P(A)/P(B)

(c) If A and B are mutually exclusive then P(A|A B) = P(A)/( P(A) + P(B) )

(d) P(A B C) = P(A|B C)P(B|C)P(C)

2. Two people each toss a fair coin n times. Find the probability that they toss the same number of heads.