Expected number of distinct faces in six die rolls

Question

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

Accepted Answer

Let X be the number of distinct faces seen. For face f \in \{1, \dots, 6\}, define If = 1 if face f appears at least once in the six rolls. Then X = \sum{f=1}^{6} If. Face f is missed by a single roll with probability 	frac{5}{6}, and missed by all six rolls with probability \left(	frac{5}{6}ight)^6, so E[If] = P(	ext{face } f 	ext{ appears}) = 1 - \left(\frac{5}{6}ight)^6 = 1 - 0.3349 = 0.6651. By linearity of expectation, E[X] = 6 \cdot \left(1 - \left(	frac{5}{6}ight)^6ight) = 6 \cdot 0.6651 \approx 3.99. So you expect to see about 4 of the 6 faces. In general, n rolls of an s…

Expected number of distinct faces in six die rolls

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