Make me a market on the min of two dice

Question

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

Accepted Answer

For the min of two independent dice, count outcomes where both dice are \ge k: P(\min \ge k) = \left(	frac{7-k}{6}ight)^2, so P(\min = k) = P(\min \ge k) - P(\min \ge k+1) = \frac{(7-k)^2 - (6-k)^2}{36} = \frac{13 - 2k}{36}. Check it is a distribution: 	frac{11+9+7+5+3+1}{36} = 	frac{36}{36} = 1. Then E[\min] = \sum{k=1}^{6} k\cdot\frac{13-2k}{36} = \frac{13\sum k - 2\sum k^2}{36} = \frac{13(21) - 2(91)}{36} = \frac{273 - 182}{36} = \frac{91}{36} \approx 2.53. A slick shortcut: \min + \max = X + Y, so E[\min] = 7 - E[\max] = 7 - 	frac{161}{36} = 	frac{91}{36}, the same answer. The mark…

Make me a market on the min of two dice

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