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Highest of $3$ dice

Expected valueMedium

$3$ fair $6$ -sided dice are rolled. What is the expected value of the highest result?

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Hint

Use P(max > k) = $1$ − P(max ≤ k).

Solution

The maximum is at most $k$ with probability $(\frac{k}{6})^{3}$ , so the expectation is the sum over $k = 0, \dots, 5$ of $1 - (\frac{k}{6})^{3}$ .
That equals $6 - \frac{0 + 1 + 8 + 27 + 64 + 125}{216} = 6 - \frac{225}{216}$ .
Hence the expectation is $6 - \frac{25}{24} = \frac{119}{24}$ .

Answer: $\frac{119}{24}$