The slowest coupon collector's problem

TL;DR

This paper extends the classical coupon collector's problem to multiple competing players, analyzing the expected times for the fastest and slowest players to complete their collections and exploring related probabilities.

Contribution

It introduces a multi-player extension of the coupon collector's problem, providing new analytical results for the expected completion times and probability distributions.

Findings

Expected number of boxes for fastest and slowest players calculated.

Probabilities of a player being the slowest or fastest analyzed.

Solutions derived using algebraic and probabilistic methods.

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, we extend the classical problem to k players who compete with one another to be the first to collect the whole collection. We 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. The solutions will be discussed from both the tractable algebraic techniques as well as the probability point of views.