The Slowest Coupon Collector’s Problem

Abstract

In the classical coupon collector’s problem, every box of breakfast cereal contains one coupon from a collection of n distinct coupons, each equally likely to appear. The goal is to find the expected number of boxes a player needs to purchase to complete the whole collection. In this work, the authors extend the classical problem to k players who compete with one another to be the first to collect the whole collection. The authors find the expected numbers of boxes required for the slowest and fastest players to finish the game. The odds of a particular player being the slowest or fastest player will also be touched upon. Using the law of total expectation, the solutions will be discussed from both the tractable recurrence relation as well as the probability point of views.