THE MULTIPLE SUBSET COUPON COLLECTING PROBLEM

Abstract

Suppose that there are n types of coupons and that each new coupon collected is type i with probability pi. Suppose, further, that there are m subsets of coupon types and that coupons are collected until all of the types of at least one of these subsets have been collected. When these subsets have no overlap, we derive expressions for the mean and variance of the number of coupons that are needed. In the general case where the subsets can overlap, we derive the mean of the number that are needed. We also note that this number is an increasing failure rate on average random variable and we present a conjecture as to a sufficient condition for it to be an increasing failure rate random variable.