Boxes needed for five different coupons

Question

Each cereal box contains one coupon numbered 1 to 5, each number equally likely and independent across boxes. You need one coupon of each number to claim a prize. How many boxes do you expect to buy?

Short answer

$137/12\approx 11.42$ . After collecting $k$ distinct numbers, a new one arrives with probability $\tfrac{5-k}{5}$ , taking $\tfrac{5}{5-k}$ boxes on average. Summing gives $5\left(1+\tfrac12+\tfrac13+\tfrac14+\tfrac15\right)$ .

Adapted from OpenQuant; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Sample Spaces, Events, and the Probability Axioms
Bond Prices, Discount Factors, and the Law of One Price
Conditional Probability, Total Probability, and Bayes' Rule

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