Boxes needed to collect five equally likely prizes

Question

Each cereal box contains one of five different toys, chosen independently and uniformly at random. What is the expected number of boxes you must open before you own at least one of every toy?

Short answer

$137/12$ . This is the coupon collector problem: after collecting $j$ distinct toys, each new box is new with probability $\tfrac{5-j}{5}$ , so the wait is geometric with mean $\tfrac{5}{5-j}$ . Summing, $5\left(1+\tfrac12+\tfrac13+\tfrac14+\tfrac15\right)=\tfrac{137}{12}\approx11.42$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Expectation, Mean, and Variance
Independence
Conditional Expectation and Variance Revisited

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