Rolls needed to see every face of a die

Question

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

Short answer

$14.7$ . This is the coupon collector problem. Once $i$ faces have been seen, the wait for a new one is geometric with success probability $\tfrac{6-i}{6}$ , so the expected total is $6\left(1+\tfrac12+\cdots+\tfrac16\right)=6\cdot\tfrac{49}{20}=14.7$ .

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

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Lessons behind this question

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

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