One of four different prizes was randomly put into each box

One of four different prizes was randomly put into each box of a cereal. If a family decided to buy this cereal until it obtained at least one of each of the four different prizes, what is the expected number of boxes of cereal that must be purchased?

Solution

This is a very common puzzle, its called coupon collection problem. Following is case with n prizes or coupons to be collected.

Let C_i be the random variable which denotes the number of coupons we buy in the stage i.

Let C be the random variable which denotes the number of coupons we buy in order to get all n coupons
Now,
C= C_1 + C_2 + ... + C_n
Note C_1 = 1;
In general, while we are during stage i , we already have with us (i-1) different coupons. So, the probability of getting a new coupon is p_i = n-(i-1)/n. Also note that each C_i is a Geometrically distributed random variable with success probability equal to p_i. So, E(C_i) = 1/p_i = n/(n-i+1).

By using Linearity of Expectations,
E(C) = E(C_1) +E(C_2) + ...+E(C_n)
= n(1/n + 1/(n-1)+ ... +1)

= 4*(1/4 + 1/3 + 1/2 + 1) = 8.3333