Three Hats

 
Three Hats

Problem statement:

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. All hats could be green, or blue, or 1 blue and 2 green, or 2 blue and 1 green.
 
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 strategy would give the best chance of success? (Hint: 100% chance of success is not possible.)

Done thinking? Click here to view the solution.

Simple strategy: Elect one person to be the guesser, the other two pass. The guesser chooses randomly “green” or “blue”. This gives them a 50% chance of winning.
 
Better strategy: If you see two blue or two green hats, then write down the opposite color, otherwise write down “pass”. It works like this (“-” means “pass”):
 
Hats: GGG, Guess: BBB, Result: Lose
Hats: GGB, Guess: –B, Result: Win
Hats: GBG, Guess: -B-, Result: Win
Hats: GBB, Guess: G–, Result: Win
Hats: BGG, Guess: B–, Result: Win
Hats: BGB, Guess: -G-, Result: Win
Hats: BBG, Guess: –G, Result: Win
Hats: BBB, Guess: GGG, Result: Lose
 
Result: 75% chance of winning!