Three players enter a room and a red or blue hat is placed on each person's head. The color of each hat is determined by a coin toss, with the outcome of one coin toss having no effect on the others. Each person can see the other players' hats but not his own. No communication of any sort is allowed, except for an initial strategy session before the game begins.
Once they have had a chance to look at the other hats, the players must simultaneously guess the color of their own hats or pass. The group shares a hypothetical $3 million prize if at least one player guesses correctly and no players guess incorrectly. One obvious strategy for the players, for instance, would be for one player to always guess "red" while the other players pass.
a. What is the expected amount of money the players win following the above strategy?
b. Suggest a different strategy and compute the expected win for it.
