The smaller of two dice

Question

You roll two fair six-sided dice and record the smaller of the two faces (if they tie, that common value). What is the expected value of this minimum?

Accepted Answer

A clean method uses the "sum of tail probabilities" identity for a positive integer variable: E[\min] = \sum{k=1}^{6} P(\min \ge k). The minimum is at least k exactly when both dice are at least k. Each die is \ge k with probability \frac{7-k}{6}, so by independence P(\min\ge k) = \left(\frac{7-k}{6}\right)^2: E[\min] = \sum{k=1}^{6}\left(\frac{7-k}{6}\right)^2 = \frac{6^2 + 5^2 + 4^2 + 3^2 + 2^2 + 1^2}{36} = \frac{91}{36} \approx 2.53. As a sanity check, the maximum has expectation \frac{161}{36}\approx 4.47, and \min + \max has mean \frac{91+161}{36} = 7, exactly the expected sum of two dic…

The smaller of two dice

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