Expected rolls to see all six die faces

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You roll a fair six-sided die repeatedly. You stop once every one of the six faces has appeared at least once. What is the expected number of rolls?

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Track how many distinct faces you have seen. With j faces already seen, a roll shows a new face with probability \frac{6 - j}{6}. The expected rolls to get the next new face satisfies ej = 1 + \frac{j}{6} ej, so ej = \frac{6}{6 - j}. Summing over j = 0,\dots,5: E = \frac66 + \frac65 + \frac64 + \frac63 + \frac62 + \frac61 = 6\Big(1 + 	frac12 + 	frac13 + 	frac14 + 	frac15 + 	frac16\Big) = 6 \cdot \frac{49}{20} = \frac{147}{10} = 14.7. Takeaway. Seeing all six faces averages 14.7 rolls; waiting for the final missing face alone costs 6 rolls on average.

Expected rolls to see all six die faces

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