32. (The Hat Problem) A game begins with a team of three players entering a room one at a time. For each player, a fair coin is tossed. If the outcome is heads, a red hat is placed on the player’s head, and if it is tails, a blue hat is placed on the player’s head. The players are allowed to communicate before the game begins to decide on a strategy. However, no communication is permitted after the game begins. Players cannot see their own hats. But each player can see the other two players’ hats. Each player is given the option to guess the color of his or her hat or to pass. The game ends when the three players simultaneously make their choices. The team wins if no player’s guess is incorrect and at least one player’s guess is correct. Obviously, the team’s goal is to develop a strategy that maximizes the probability of winning. A trivial strategy for the team would be for two of its players to pass and the third player to guess red or blue as he or she wishes. This gives the team a 50% chance to win. Can you think of a strategy that improves the chances of the team winning?