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Three players enter a room and a red or blue hat is placed on each person’s head. The color of each hat is determined by independent coin tosses. Each person can see the other player’s hats but not his/her own. No communication of any sort is allowed, except for an initial strategy session before the game begins. Once they have had a chance to look at the other hats, the players must simultaneously guess the color of their own hats or pass. The group shares a prize if at least one player guesses correctly and no other player guesses incorrectly. Find a strategy for the group that maximizes its chances of winning the prize.

i have realised that since only one person needs to guess and with the greater number of people who guess comes the greater the chances of someone guessing incorectly, that only one person should guess. also the color of the others hats will not help the person who is guessing determine their own hat color. therefore the best strategy that i can come up with is to have one predetermined person randomly guess red or blue and the other two pass (50% of being right and winning). but winning only half the time is not a good startegy.

if you can think of a better, mor effective way to do the proble or can figure out a better way if the problem was slightly changed in any way (i.e. the three people would not guess simultaneously), please help me out