Make me a market on the max of two dice

Question

Two fair six-sided dice are rolled. A contract pays \max(X, Y) in dollars. Compute the fair value exactly and make a two-sided market.

Accepted Answer

For the max of two independent dice, P(\max \le k) = \left(	frac{k}{6}ight)^2, so P(\max = k) = \frac{k^2 - (k-1)^2}{36} = \frac{2k-1}{36}. Then E[\max] = \sum{k=1}^{6} k \cdot \frac{2k-1}{36} = \frac{\sum{k=1}^{6}(2k^2 - k)}{36} = \frac{2(91) - 21}{36} = \frac{161}{36} \approx 4.47. (Check the distribution sums to one: 	frac{1+3+5+7+9+11}{36} = 	frac{36}{36}.) A slick alternative: \max + \min = X + Y, so E[\max] = 7 - E[\min], and by the same counting E[\min] = 	frac{91}{36}; indeed 	frac{161}{36} + 	frac{91}{36} = 	frac{252}{36} = 7. The market. Fair \approx 4.47. Quote e.g. 4.3 bi…

Make me a market on the max of two dice

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