# Probability: A and B play a game with two six-sided dice. Each die has some red faces and some blue faces. The two dice are thrown simultaneously.

A and B play a game with two six-sided dice. Each die has some red faces and some blue faces. The two dice are thrown simultaneously. If the top face of each die is the same color, A wins, and if they are different colors, B wins. One die has ﬁve red faces and one blue face. How many red faces does the other die have if both players have the same chance of winning?

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If there are $$r$$ red and $$b$$ blue faces on the second die, then A will win with probability $$\frac56\cdot\frac r6+\frac 16\cdot\frac b6.$$ This is supposed to be $$\frac12$$, and of course $$r+b=6$$. Solve these equations.