Rolls needed to see every face of a die

Question

A fair six-sided die is rolled until each of the six faces has appeared at least once. What is the expected number of rolls?

Short answer

$14.7$ . This is the coupon collector problem. Once $k$ faces have been seen, a new one appears with probability $\tfrac{6-k}{6}$ , so the wait is geometric with mean $\tfrac{6}{6-k}$ . Summing, $6\left(1+\tfrac12+\dots+\tfrac16\right)=6\cdot\tfrac{49}{20}=14.7$ .

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Lessons behind this question

Expectation, Mean, and Variance
Counting and Combinatorics
Conditional Expectation and Variance Revisited

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