Three people are trying to win the following game as a team: each of them is put on a hat of either red or blue with i. i.d probability of 1/2. (i. e. equal chance of being red and blue, and what's put on one person doesn't affect what are on the other people.) each one can only see the other people's hats, but not his own. he has to guess the color of his own hat by writing down either "red", "blue", or "don't know". after all three people write down their guesses, they would win if:
1. at least one of them guessed right, and
2. none of them guessed wrong. note: "guessed right" is defined as guessing a color that is the color of the hat. "guessed wrong" is defined as guessing a color that is not the color of the hat. it's neither "right" nor "wrong" if "don't know" is guessed.
those three people can discuss a strategy before the hats are put on their heads. after the hats are on, they can't communicate to each other including seeing other's guess. what strategy would give them the best chance of winning and what's the probability of winning under that strategy?