Three dice, expected value of the smallest one

Question

You roll three fair six-sided dice and look at the smallest of the three faces. On average, what is the value of the smallest die?

Accepted Answer

There are 6 	imes 6 	imes 6 = 216 equally likely rolls. Find how often the smallest die equals each value k. How often is the smallest die exactly k? The smallest die is k or more when all three dice are k or more. The faces from k to 6 number 7-k, so all three being at least k happens in (7-k)^3 ways. Then the smallest is exactly k in (7-k)^3 - (6-k)^3 ways. For k = 1 through 6 that gives 91, 61, 37, 19, 7, 1 ways, which add to 216. Average. Multiply each k by its count and add: 1(91) + 2(61) + 3(37) + 4(19) + 5(7) + 6(1) = 91 + 122 + 111 + 76 + 35 + 6 = 441. Divide by 216: EV = \frac{441}…

Three dice, expected value of the smallest one

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