Problem 10. Hats
Three people enter a room and someone randomly puts red or blue hats on each of them. Everyone sees the color of the hats of the other two people, but does not see the color of his own hat.
Each person either makes a guess on the color of his own hat and writes the color down on a piece of paper or leaves the paper blank. All the papers then are given to a judge. If at least one of the guesses is correct and none of them is wrong, they will win a prize. What strategy guarantees the highest probability of winning? How likely is chance of winning with that strategy?
More details:
They can agree on a strategy before they enter the room, but no information should be exchanged between them inside the room.
Each of the 8 combinations of hat colors will realize with probability \(1/8\).
Link to the problem on Twitter: https://twitter.com/Riazi_Cafe/status/1675054034577952768
. With the right strategy, they can guarantee a \(75\%\) chance of winning.
Two examples of strategies that guarantee this possibility:
Strategy 1. If the color of the hats of the other people are the same, write the opposite color on your paper. Otherwise leave it blank. In this case, the players will win in all cases except for the ones in which all three hat colors are the same.
Strategy 2. First person: If the two other people have the same color on their hats, write the same color on your paper. Otherwise, leave it blank. 2nd and 3rd person: If the two other people have the same colors on their hats, leave your paper blank, otherwise write down the color of the hat of the first person on the paper.