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**mashplum**- Replies: 2

Three people enter a room and have a green or blue hat placed on their head. They cannot see their own hat, but can see the other hats.

The color of each hat is purely random. They could all be green, or blue, or any combination of green and blue.

They need to guess their own hat color by writing it on a piece of paper, or they can write "pass".

They cannot communicate with each other in any way once the game starts. But they can have a strategy meeting before the game.

If at least one of them guesses correctly they win $50,000 each, but if anyone guess incorrectly they all get nothing.

What is the best strategy?

The given solution offers a 75% chance at winning. Here is a 100% method:

Before the game the people agree that if they see a blue and a green hat, they will write pass. If they see a match, they will pause. If everyone pauses, they must all have the same color and they can all safely guess that color. If two people immediately write pass, the third knows to guess the opposite of what the other two are wearing.

If your initial choice happened to be the car, then switching would always get you a goat. But if your initial choice was a goat then switching will get you a car. Switching will always get you the opposite of whatever your initial choice was. If you agree with that last statement the solution is obvious. Everyone agrees that your initial choice has only a 1/3 probability of being right and a 2/3 probability of being wrong, so switching would have to be the opposite of that. In other words, would you rather bet that your first choice was a car or bet that it was a goat?

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