# Hats

Probability Level 4

A group of six people are playing a game. They stand in a line, and each can only look forward. A rabbit then randomly places a red or blue hat on each person’s head. At this moment, people can see only the hats in front of them, and not their own or those behind them. Now each person places a bet on what color his/her hat is, without knowing anyone else’s bet. The bet can be any nonnegative amount of money; if it is correct, the group gains that much money, but if it is incorrect, the group loses that much money. The group wins the game if, at the end, they have gained money (however small the amount). The six people are given an opportunity to devise a strategy before playing. If they use an optimal strategy, what is the maximum probability they will win the game?

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