Expected value of the minimum of dice

Question

You roll n fair six-sided dice and record the smallest value that appears. What is the expected value of this minimum? Work out n = 2 exactly, then give the general formula.

Accepted Answer

Let m be the minimum of n i.i.d. fair dice. Tail event for the min. The minimum is at least k iff all dice show k or higher. There are 7 - k faces from k to 6, so P(m \ge k) = \left(\frac{7 - k}{6}\right)^{n}, \qquad k = 1, \dots, 6. Tail-sum formula. E[m] = \sum{k=1}^{6} P(m \ge k) = \sum{k=1}^{6}\left(\frac{7 - k}{6}\right)^{n} = \frac{1^n + 2^n + \cdots + 6^n}{6^{n}}. Case n = 2: E[m] = \frac{1 + 4 + 9 + 16 + 25 + 36}{36} = \frac{91}{36} \approx 2.528. Symmetry check. By the reflection k \mapsto 7 - k, the minimum of the dice mirrors the maximum, so E[\min] + E[\max] = 7. With E[\max] = 16…

Expected value of the minimum of dice

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