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.
Assume an equal chance of getting a red or a blue hat, separately for each participants.
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 submit in writing their guesses, they would jointly win if:
1. At least one of them guessed right,
2. None of them guessed wrong .
"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 submitted as an answer.
Those three people can establish a joint strategy before the hats are put on their heads.
After the hats are on, they can neither communicate to each other nor see other guesses.
What strategy would give them the best chance of winning and what's the probability of winning under that strategy?
Source: Allegedly posted in the elevator of UC Berkeley Math department.