Question

Each box of cereals contains a coupon. If there are four kinds of coupons, how many boxes have to be bought on average to obtain all four coupons? Remark. This is problem is known as Coupon Collector' Problem.

Answer 1

The no of box until the first coupon appears is 1. After that, the random time until a second (different) coupon appears is geometrically distributed with parameter of success 3/4 hence with mean 4/3(recall that the mean of a geometrically distributed random variable is the inverse of its parameter). After that, the random no of boxes until a third (different) coupon appears is geometrically distributed with parameter of success 2/4, hence with mean 4/2. And after that, the random no of boxes until a fourth (different) coupon appears is geometrically distributed with parameter of success 1/4, hence with mean 4/1. This shows that the mean total no of boxes to get all four results is

1 + 4/3 + 4/2 + 4/1= 25/3 = 8.3333