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Largest die payoff

Market makingHard

$3$ fair $6$ -sided dice are rolled independently. A contract pays, in dollars, the largest of the numbers. What is its fair value in dollars? Give an exact fraction.

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Hint

Use E[M] as the sum over k of P(M ≥ k), or count outcomes by maximum.

Solution

The maximum is at most m with probability $(\frac{m}{6})^{3}$ , so it is at least k with probability $1 - (\frac{k - 1}{6})^{3}$ .
Summing for $k$ from $1$ to $6$ gives $6 - \frac{1 + 8 + 27 + 64 + 125}{216} = 6 - \frac{225}{216}$ .
The fair value is $\frac{119}{24}$ .

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

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