Discrete Math question. Please show all steps and proofs! Thanks. Three people P

Question

Discrete Math question. Please show all steps and proofs! Thanks.

Three people P, Pr, and are a room. person has a bag containing one red hat and one blue hat. Each person chooses a uniformly random hat from her bag and puts it on her head. Afterwards, the lights are turned on. Each person does not know the color of her hat, but can see the colors of the other two hats. Each person Pi can do one of the following: Person P announces "my hat is red" Person Fi announces "my hat is blue" Person Fi says "I pass". The game is a success if at least one person announces the correct color of her hat and no person announces the wrong color of her hat. (If a person passes, then she does not announce any color. Assume person announces my hat is red" and both Pr and PB pass. Define the event A "the game is a success Determine Pr(A). Show your work.

Explanation / Answer

Solution:- Three persons P1,P2,P3 are in a dark room .Each is person is having a bag which contains one blue and one red hat.

Let us suppose that P1 announces that my hat is red,and both P2 and P3 pass.

A=The game is success,

To determine the Pr(A),

If P1 observes that P2 and P3 both have same color hats, then P1 announces the color which was on P2 and P3 posses.If P1's assumption was right,then P1 get pass, the game is success.

If P2, P3 both are having different colors, then it is difficult to predict P1 to announce the color,at that time he says nothing.

So, P2 by taking some interval of time,he observes that P1 says nothing and with her knowledge ,she can see the color of P3's hat and announces what color she may posses on her head.If she observes red color on P3's head then, she announces that her hat is in blue color.

Then, P2 will get success.

One person know that there is only two hats of each color, So, if P1 observes that P2 and P3 have hats of the same color, P1 would conclude that, his own hat may be the opposite color.If P2 and P3 are having different colors, then P1 cannot say anything.

The person P2, can predict what is her hat on her head,This is because if P1 announces a color, that is if P1 says nothing , then P2see the color on P3 head and predict that the opposite color is her hat.

By this way we can determine the success of Pr(A) and Pr(B).

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