Expected rolls to see all six faces

Question

You roll a fair six-sided die repeatedly. What is the expected number of rolls until you have seen every face at least once?

Accepted Answer

This is the coupon collector problem with n = 6. Decompose the total time as T = T1 + T2 + \cdots + T6, where Tk is the number of rolls to get the k-th new face after k-1 distinct faces have been seen. While you hold k-1 distinct faces, each roll shows a new face with probability pk = \frac{6-(k-1)}{6}. So Tk is geometric with mean 1/pk = \frac{6}{7-k}. By linearity of expectation, E[T] = \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\right) = 6 H6. Since H6 = \frac{49}{20}, E[T] = 6 \cdot \frac{49…

Expected rolls to see all six faces

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