Expected Maximum

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Roll two fair dice. What is the expected value of the larger of the two results (counting ties as that common value)?

Accepted Answer

The cleanest route is the tail-sum formula for non-negative integer variables: mathbbE[M] = sum_k=1^6 P(M ge k) = sum_k=1^6left[1 - P(textboth < k)right] = sum_k=1^6left[1 - left(frack-16right)^2right]. = 6 - frac0+1+4+9+16+2536 = 6 - frac5536 = frac16136 approx 4.472. Symmetry check (great for catching errors): min + max = X + Y, so mathbbE[min] = 7 - frac16136 = frac9136 approx 2.528. The max sits about as far above 3.5 as the min sits below, exactly as it must. The tail formula mathbbE[X] = sum_kge1 P(X ge k) is worth making reflexive; it routinely turns order-statistic questions into two-line computations.

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