Two dice, expected value of the larger one

Question

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

Accepted Answer

There are 6 	imes 6 = 36 equally likely rolls. A neat way to find the average of the larger die is to first find, for each value k, how often the larger die equals k. How often is the larger die exactly k? The larger die is k or less when both dice are k or less, which happens in k 	imes k = k^2 ways. So the larger die is exactly k in k^2 - (k-1)^2 = 2k - 1 ways. Check: that gives 1, 3, 5, 7, 9, 11 ways for k = 1 through 6, which add to 36, matching all rolls. Average. Multiply each value k by its count 2k-1, add, and divide by 36: \sum{k=1}^{6} k(2k-1) = 1(1) + 2(3) + 3(5) + 4(7) + 5(9) + …

Two dice, expected value of the larger one

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