In: Statistics and Probability
Three people are brought into a room. A hat is placed on each person’s head. The hat is equally likely to be Red or Blue. (So each of the 8 possibilities is equally likely.) Each person sees the colors of the other people’s hats, but not their own. Each person, without communication, writes down one of the following: "My hat is red", "My hat is blue" or "Pass". All three people will be put in jail unless (a) at least one of them doesn’t pass, and (b) everyone who doesn’t pass is right about his/her own hat color. Importantly, they can agree ahead of time on a strategy, with the hopes of not going to prison.
(a) What is the probability that they are not sent to prison if each person guesses the color of his/her own hat?
(b) What is the probability that they are not sent to prison if two of them pass and 1 of them guesses?
(c) What is the probability that they are not sent to prison if they use the following strategy: Each person looks at the other two hats. If they are both blue, then the person guesses red. If they are both red, then the person guesses blue. If they are different, the person passes.
If everyone of the three person guesses the color of his hat, It is likely that each person is taking a 50% chance. There are two colors with equally like chance of 1/2 in a guess like tossing a coin with blue and red color on it. There are 8 outcomes, there is chance of 4 will be correct, so 4/8.
Two persons pass, and third guess the color of his hat, he thinks of the possibilities after seeing the colors of the two persons. Since there are two colors Red and Blue, he can reject the two possibilities that is one with all three blue or all three red. Now he can gues one of the six other possibilities of the hat colors, so he has chance of guessing the right color by 6/8 = 0.75=75%.
First two hats of same colors occur in 2 ways, One is Blue-Blue; another one is Red - Red. If one guesses the other color, he can be right, It occurs in 2 cases with a probability 2/8=25%, again this happens one of the six winning possibilities of the event. So there 6/8 chance of wiining it. So there is 75% chance that they will not be sent to prison