Rolls of a die to see every face at least once

Question

You roll a fair six-sided die repeatedly until every one of the 6 faces has appeared at least once. On average, how many rolls does that take?

Accepted Answer

This is coupon collecting with 6 equally likely types. When you have already seen k distinct faces, a roll shows a new face with probability \frac{6 - k}{6}, so that stage averages \frac{6}{6 - k} rolls. Summing k = 0, \ldots, 5: E = \frac{6}{6} + \frac{6}{5} + \frac{6}{4} + \frac{6}{3} + \frac{6}{2} + \frac{6}{1} = 6\left(1 + 	frac12 + 	frac13 + 	frac14 + 	frac15 + 	frac16ight). The harmonic sum H6 = \frac{49}{20}, so E = 6 \cdot \frac{49}{20} = \frac{294}{20} = 14.7 	ext{ rolls.} Takeaway. Seeing all 6 faces averages 6 H6 = 14.7 rolls; the final missing face alone costs about 6 of t…

Rolls of a die to see every face at least once

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