Question

Three players enter a room. As each player enters, a coin is flipped (independently of the other players) and either a red or blue hat is placed on that player's head. The players can not see the colors of their own hats, only the colors of the other two players hats. Once everyone is in the room, each of the players will simultaneously guess his hat color or pass. If at least one player guesses correctly and none guess incorrect, the players will win a cash prize. There is no communication between the players once they enter the room, but they are allowed to discuss a strategy prior to entering. The naive strategy would be for one player to just guess red and the other two pass, giving a probability of winning 1/2. Is there a strategy that gives a higher probability? Hint: there is a strategy that yields 75%

Answer 1

There are 8 possible arrangements of the hats, each having the same probability of occuring (assuming a fair coin).
BBB BBR BRB RBB RRB RBR BRR RRR
The agreed strategy would be: If the other two players are wearing the same color, then guess the opposite color. Otherwise pass. This strategy fails for RRR and BBB only, but wins in the other 6 cases.