1 Some Interesting Problems In Probability Prof. Alagar Rangan Dept of Industrial Engineering Eastern Mediterranean University Dept of Industrial Engineering Eastern Mediterranean University North Cyprus North Cyprus

Problem 1:Prisoner’s Dilemma 2 Three prisoners, A, B and C, with apparently equally good records have applied for parole. The parole board has decided to release two of the three, and the prisoners know this but not which two. A warder friend of prisoner A knows who are to be released. Prisoner A realizes that it would be unethical to ask the warder if he, A, is to be released, but thinks of asking for the name of one prisoner other than himself who is to be released. He thinks that before he asks, his chances of release are 2/3. He thinks that if the warder says “B will be released,” his own chances have now gone down to ½, because either A and B or B and C are to be released. And so A decides not to reduce his chances by asking. However, A is mistaken in his calculations. Three prisoners, A, B and C, with apparently equally good records have applied for parole. The parole board has decided to release two of the three, and the prisoners know this but not which two. A warder friend of prisoner A knows who are to be released. Prisoner A realizes that it would be unethical to ask the warder if he, A, is to be released, but thinks of asking for the name of one prisoner other than himself who is to be released. He thinks that before he asks, his chances of release are 2/3. He thinks that if the warder says “B will be released,” his own chances have now gone down to ½, because either A and B or B and C are to be released. And so A decides not to reduce his chances by asking. However, A is mistaken in his calculations.

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Problem 2:Meeting of Lovers A gentleman and his girl friend, make an appointment to meet on a certain day at a certain place between 4 P. M. to 5 P. M. at random points of time and agree that each is to wait not more than 15 minutes for the other. Assuming that each is independently equally likely to arrive at any time during the hour, find the probability that they meet. A gentleman and his girl friend, make an appointment to meet on a certain day at a certain place between 4 P. M. to 5 P. M. at random points of time and agree that each is to wait not more than 15 minutes for the other. Assuming that each is independently equally likely to arrive at any time during the hour, find the probability that they meet. 4

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Problem 3:The Flippant Juror 6

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Problem 4:Random Quadratic Equations What is the probability that the quadratic equation What is the probability that the quadratic equation has real roots? has real roots? 8

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Problem 5:Travel To Freedom 10

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Problem 6:The Matching Problem At a party n people put their hats in the center of a room where the hats are mixed together. Each person then randomly selects one. Probability of no matches? At a party n people put their hats in the center of a room where the hats are mixed together. Each person then randomly selects one. Probability of no matches? 12

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Problem 7:Making A Triangle 15

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