# Probability: Optimal Choice of Dice

A game is played in which one of two players chooses a dice from a set of three die. After the first player chooses, the second player chooses a dice from the remaining two that are left. Then the two role their dice simultaneously. The player that turns up the highest number wins. In this game, however, although the dice are fair in the sense that any side is equally likely to come up, the numbers on the face of the three dice differ. Dice 1 has the following six numbers on its face: {5,7,8,9,10, 18}; dice 2 has {15, 16, 17, 2, 3, 4}; and dice 3 has {1,6,11,12,13,14}. Find player 2's optimal choice of dice given any choice of dice by player 1. Which dice should player 1 choose? The problem is just one of finding the probability of winning given any two dice that are chosen.

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#### Solution Preview

1. Let us call the three dice as A, B and C in the given order.

Case 1: The first player chooses A.

Then the second player's expectation if she chooses B = (1/6)(15 + 16 + 17 + 2 + 3 + 4) = 9.5

The second player's expectation if she chooses C = (1/6)(1 + 6 + 11 + 12 + 13 + ...

#### Solution Summary

This solution is a review on finding the probability of winning two die in a game from a set of three die.