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Hats revisited (Posted on 2013-12-19) Difficulty: 2 of 5
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.

No Solution Yet Submitted by Ady TZIDON    
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Solution Logicians don't discuss strategy (spoiler?) | Comment 1 of 6
They can achieve 50% success if two agree in advance to write "Don't know" and the third guesses.

They can achieve 75% success if each agrees to write "Don't know" when they see different colors on the heads of the other two, and the opposite color when they see matching colors on the head of the other two.  This fails when all three hats are the same (which occurs with probability 25%), as all three will guess wrong.  It succeeds the other 75% of the time, as the odd man-out will correctly guess his hat color and the other two will write "Don't know".  

If they were all logicians schooled in probability, they would not even need to discuss this strategy in advance.  Each could come up with the winning strategy (it's only D2, after all) and each would do the right thing, secure in the knowledge that their fellow logicians will do the same.  This assumes that I have correctly arrived at the winning strategy, which is not clear yet. 

  Posted by Steve Herman on 2013-12-19 12:28:22
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