Expected number of distinct faces in six rolls

Question

You roll a fair six-sided die 6 times. What is the expected number of distinct faces that appear?

Accepted Answer

Let Ik indicate that face k appears at least once in the six rolls, so the number of distinct faces is D = \sum{k=1}^{6} Ik. Face k is missed by all six rolls with probability \left(\frac{5}{6}\right)^6, so E[Ik] = 1 - \left(\frac{5}{6}\right)^6. By linearity of expectation (dependence between faces is irrelevant): E[D] = 6\left(1 - \left(\frac{5}{6}\right)^6\right) = 6\left(1 - \frac{15625}{46656}\right) = 6 \cdot \frac{31031}{46656} = \frac{31031}{7776} \approx 3.99. Only about 4 distinct faces, not 6, collisions are the norm, not the exception. (Compare: getting all six faces in exactly si…

Expected number of distinct faces in six rolls

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