Part of a well-known puzzle involves three people entering a room. As each person enters, at random either a red of a blue hat is placed on the person's head. The probability that an individual receives a red hat is .

No person can see the color of the other two persons' hat. The three will split a prize if at least one person guesses the color of their own hat correctly and no one guesses incorrectly. A person may decide to pass rather than to guess. The three people are not allowed to confer with one another once the hats have been placed on their heads, but they can agree on a strategy prior to entering the room. At the risk of spoiling the puzzle, one strategy the players may follow instructs a player to pass if they see the other two persons wearing mis-matched hats and to guess the opposite color if their friends are wearing matching hats. Why is this a good strategy and what is the probability of winning the game?