Three dice, expected value of the largest one

Question

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

Accepted Answer

There are 6 	imes 6 	imes 6 = 216 equally likely rolls. Find how often the largest die equals each value k. How often is the largest die exactly k? The largest die is k or less when all three dice are k or less, which happens in k^3 ways. So the largest is exactly k in k^3 - (k-1)^3 ways. For k = 1 through 6 that gives 1, 7, 19, 37, 61, 91 ways, which add to 216. Average. Multiply each k by its count and add: 1(1) + 2(7) + 3(19) + 4(37) + 5(61) + 6(91) = 1 + 14 + 57 + 148 + 305 + 546 = 1071. Divide by 216: EV = \frac{1071}{216} = \frac{119}{24} \approx 4.96. With three dice instead of two, …

Three dice, expected value of the largest one

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