# How many boxes of cereal do you expect to have to buy to get all N toys?

Suppose 𝑵 different toys are offered in boxes of a favorite brand of cereal. You want to collect all the different toys. How many boxes of cereal do you expect to have to buy to get all 𝑵 toys? Let the random variable 𝑿 be the number of boxes of cereal you need to buy to get each toy at least once. Also, assume it is equally likely that any one of the 𝑵 toys will be in each box of cereal.

a. Show that $$𝑿 = 𝑿_𝟎 + 𝑿_𝟏 + 𝑿_𝟐 + ⋯ + 𝑿_{𝑵−𝟏}$$ , where $$𝑿_𝒌$$ has a geometric distribution with probability $$\frac{𝑵−𝒌}{𝑵}$$.

Hint: consider the random variable which is the number of boxes needed to get the 𝒏𝒕𝒉 toy after getting 𝒏 − 𝟏 of all of them.

b. Find 𝑬(𝑿). The correct answer indicates that to get all eight toys offered with the current MacDonalds Happy Meal, you would, under the assumptions of the calculation, expect to buy about $$22$$ Happy Meals.

I was able to calculate part b as $$E(X) = N(\frac{1}{N} + \frac{1}{N-1} + ... + \frac{1}{N - (N-1)})$$ but I can't figure out how to start part a.

• There is literature on this. It's called the Coupon Collector's Problem. – Gerry Myerson Feb 12 at 2:58
• I found information on the coupon collector's problem before, but I haven't been able to properly link it to what part a of the question is asking me to do. I was able to use it to solve part b, but whatever material I read on the problem doesn't clear up part a at all for me. – hmtkd Feb 12 at 3:16