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7 people receive either a black or a white ball. They can only see the color of the others balls, but not their own. Both of the colors are equally likely. They play as a team a game of guessing their own ball color.

With which strategy all of the 7 people answer correctly; and which probability of success does this strategy have?

(Sidenotes: The strategy should be made before they received the balls. And: The 7 people cannot communicate anything once they received the balls.)

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Here is an optimal strategy:

  • If you see an odd number of black hats, guess black.

  • If you see an even number of black hats, guess white.

As long as the total number of black hats is even, everyone will guess correctly. This occurs with probability 50%.

You cannot do any better, because no matter what strategy people use, each person will be wrong half the time on average. This is because each person’s hat is independent of their guess, as the guess depends on the other hats only, and all hats are independent. No matter what they guess, the probability their hat is the same as their guess is 50%.

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