Hats
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?
Image credit: Wikipedia LKMerzigWadem
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1 solution
You totally stole this question from the mini-test in our class :P And just saying, you got the right answer by luck, your process for getting it was completely wrong.
The main idea is this: every person can see everything that the person in front of them sees, so if they come up with a plan, they can deduce what the people in front of them guessed.
To put this into use, have the first person bet one dollar. If the people behind his sees that he guessed right, with a 2 1 chance, then they all bet 0 and they win. If he guessed wrong, then the next person bets two dollars. If everyone sees that he guessed right, they all bet 0 and they win. If he bets wrong, then the process keeps on going.
Summing up all the probabilities, we get that the final probability is 6 4 6 3 .