Three players enter a room and a red or blue hat is placed on each person\'s hea
ID: 3272588 • Letter: T
Question
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 a coin toss, with the outcome of one coin toss having no effect on the others. Each person can see the other players' hats but not his 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 hypothetical $3 million prize if at least one player guesses correctly and no players guess incorrectly. One obvious strategy for the players, for instance, would be for one player to always guess "red" while the other players pass. (a) What is the expected amount of money the players win following the above strategy? (b) Suggest a different strategy and compute the expected win for it.Explanation / Answer
A) P(the player who guesses will be wearing a red) = 0.5
So, expected outcome = 0.5x$3million = $1.5 million
B) P(3 reds) = 1/2 x 1/2 x 1/2 = 1/8
P(2 reds and 1 white) = 3 x 1/2 x1/2 x1/2 = 3/8
P(1 red and 2 whites) = 3/8
P(3 whites) = 1/8
Always, there will be at least 2 people having same colour hats. So, they should make the strategy that, if someone see two same colours on others, he shoud shout the opposite colour.
So, if all have same colours, then they all will shout together the opposite colour and they will lose money.
But, if its 2 whites and 1 red or 2 reds and 1 white, the person who see 2 same colour on others will only shout his own colour correctly. Others will see opposite colours and will pass. So, in such cases, they will win.
Expected amount to be won = 0x(1/8 + 1/8) + (3/8+3/8)x(3 million)
= $2.25 million
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