You and two friends are on a game show. Each of you enters the studio with...

You and two friends are on a game show. Each of you enters the studio with a hat on your head. There are two colors of hats: red and blue. They are assigned randomly, so each person has a 50% chance of a red hat and a 50% chance of a blue hat. Each person can see the hats of the two other people, but they can't see their own hat. Each person can either try to guess the color of his or her own hat or pass. All three of you do it simultaneously, so there is no way to base your guess on the guesses of your friends. If nobody guesses incorrectly and at least one person guesses correctly, then you all share a big prize. Otherwise you all lose. If everybody randomly guesses and nobody passes what is the probability that you will win?

Since you are all friends, and you all know how the game works, you develop a strategy with your friends before the show to maximise your probability of winning. (Note: if your strategy requires a particular person to speak regardless of what hat colours you all see then that person is you. In cases where both colours are equally attractive then you will pick red.)
When you walk onto the stage you can see that both of your friends have blue hats on.
What colour should you guess?
(a) Red (b) Blue (c) Stay silent

When you walk onto the stage you can see that one of your friends has a red hat on and the other friend has a blue hat on. What colour should you guess? (
a) Red (b) Blue (c) Stay Silent
What is the probability that you will win using the best strategy?

Expert Solution

Chance for winning is 75%

Explanation:

Totally there will 8 outcomes for 3 friends and 2 colored hats. That is, for example,

From the above outcomes, there has to be two hats of same color and other is different.

From above majority minority classification, One can say how many wearing which color.

For example, if you see both a red and blue hat, then you must be wearing the majority color (which could be red or blue). If you see two blue or two red hats, then you must be wearing the minority color, which will be the opposite color of what you see. The following are the six choices for you to choose:

The idea is to get yourself with the minority hat color to guess and force the other people to win.
So the strategy is:
If you see both red and blue hat, then “win”
If you see two red hats, then guess “blue” as yours
If you see two blue hats, then guess “red” as yours

The following is the players guess:

So the group wins in six of eight possible outcomes which is 75% of chance.

b) When you walk on to the stage and see both wearing blue hats, then guess Red hat is yours.

c) The probability of win this strategy is 75% if you see both red and blue hats on your friends.

milcah answered 2 months ago

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