Expected spread between the two dice

Question

You roll two fair six-sided dice and look at the range: the larger face minus the smaller face (zero if they match). What is the expected value of this range?

Accepted Answer

By linearity of expectation, even though max and min are dependent, E[	ext{range}] = E[\max] - E[\min]. For two fair dice the two order statistics are known: E[\max] = \frac{161}{36} \approx 4.472, \qquad E[\min] = \frac{91}{36} \approx 2.528, which correctly sum to 7 (since \max + \min = X1 + X2, whose expectation is 3.5 + 3.5 = 7). Subtracting, E[	ext{range}] = \frac{161}{36} - \frac{91}{36} = \frac{70}{36} = \frac{35}{18} \approx 1.944. Direct check. The range is 0 only on the 6 doubles, probability 6/36, and is more likely to be small than large. A full enumeration of X1 - X2 gives E = …

Expected spread between the two dice

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