Two dice, expected value of the smaller one

Question

You roll two fair six-sided dice and look at the smaller of the two faces (if they tie, that shared value is the smaller). On average, what is the value of the smaller die?

Accepted Answer

There are 6 	imes 6 = 36 equally likely rolls. Find how often the smaller die equals each value k. How often is the smaller die exactly k? The smaller die is k or more when both dice are k or more. The faces from k to 6 number 7-k, so both being at least k happens in (7-k)^2 ways. Then the smaller die is exactly k in (7-k)^2 - (6-k)^2 = 13 - 2k ways. Check: for k = 1 through 6 that is 11, 9, 7, 5, 3, 1, which add to 36. Average. Multiply each k by its count 13 - 2k, add, divide by 36: \sum{k=1}^{6} k(13-2k) = 1(11) + 2(9) + 3(7) + 4(5) + 5(3) + 6(1). That is 11 + 18 + 21 + 20 + 15 + 6 = 91. …

Two dice, expected value of the smaller one

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