Expected value of the largest of three dice

Question

You roll three fair six-sided dice and record the largest number showing. What is the expected value of that largest number?

Accepted Answer

A clean way to get the expected value of a maximum is to add up the probabilities that it is "at least v" across every level v. For any positive whole-valued quantity, E[	ext{max}] = \sum{v \ge 1} P(	ext{max} \ge v). The largest of three dice is at least v unless all three dice come up below v. Each die lands below v with probability \frac{v-1}{6}, so P(	ext{max} \ge v) = 1 - \left(\frac{v-1}{6}ight)^3. Summing over v = 1 to 6: E[	ext{max}] = 1 + 	frac{215}{216} + 	frac{208}{216} + 	frac{189}{216} + 	frac{152}{216} + 	frac{91}{216} = \frac{1071}{216} = \frac{119}{24} \approx 4.958.…

Expected value of the largest of three dice

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