Consider a deck consisting of the thirteen diamonds cards which we order from ac

Question

Consider a deck consisting of the thirteen diamonds cards which we order from ace (the lowest) to king (the highest). Three players are each given one of these 13 cards.

(a) Suppose the first player has the card x. Compute the probability that he has the highest card among the three players for all values of x.

(b) Suppose the second player has the card y. She now asks the first player whether she thinks that she has the highest card among the three. The first player, being a math major, will say YES if the probability that she has the highest card exceeds 1/2 and NO if it is below 1/2 . If the first player says YES, what is the probability that the second player has the highest card among the three players for all values of y.

(c) Suppose the third player has the card z. The third player which heard the answer of the first player to the second player now asks the second player if she thinks she had the highest card among the three. If the second player says YES compute the probability that the second player has the highest card among the three players for all values of z.

Explanation / Answer

(a) Suppose the first player has the card x. Compute the probability that he has the highest card among the three players for all values of x.

probability that he has the highest = 1/13

(b) Suppose the second player has the card y. She now asks the first player whether she thinks that she has the highest card among the three. The first player, being a math major, will say YES if the probability that she has the highest card exceeds 1/2 and NO if it is below 1/2 . If the first player says YES, what is the probability that the second player has the highest card among the three players for all values of y.

Probability ( Y has higest|X says Yes) = P( Y has highest and X says yes)/P(X says Yes) = (1/12)/(12/13)=13/144

(c) Suppose the third player has the card z. The third player which heard the answer of the first player to the second player now asks the second player if she thinks she had the highest card among the three. If the second player says YES compute the probability that the second player has the highest card among the three players for all values of z.

Probability ( Y has higest|X RESPONDS) =Probability ( Y has higest|X says Yes+Probability) ( Y has higest|X says nO) =13/144

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