59. Suppose each new coupon collected is, independent of the past, a type i coupon with probability pi . A total of n coupons is to be collected. Let Ai be the event that there is at least one type i in this set. For i = j, compute P(AiAj) by

(a) conditioning on Ni , the number of type i coupons in the set of n coupons;

(b) conditioning on Fi , the first time a type i coupon is collected;

(c) using the identity P(Ai ∪ Aj) = P(Ai) + P(Aj) − P(AiAj).