Sol
44. Winning an Unfair Game A game consists of a sequence of plays; on each play either you or your opponent scores a point, you with probability p (less than )), he with probability 1 - p. The number of plays is to be even-2 or 4 or 6 and so on. To win the game you must get more than half the points. You know p, say 0.45, and you get a prize if you win. You get to choose in advance the number of plays. How many do you choose?36. Gambler's Ruin Player M has $1, and Player / has $2. Each play gives one of the players $1 from the other. Player M is enough better than Player / that he wins $ of the plays. They play until one is bankrupt What is the chance that Player M wins?3%. Birthday Holidays Labor laws in Erewhon requirc factory owners to live every worker a holiday whenever one of them has a birthday and to hire without discrimination on grounds of birthdays. Except for these holidays they work a SSS-day year. The owners want to maximize the expected total number n!" man-days worked per year in a factory. How many workers do factories have in Ercwhon '1 28. Catching the Greedy Counterfeiter The king's minter boxes his coins , to a box. Each box contains m false coins. The king suspects the minter and randomly draws I coin from each of n boxes and has these tested. What is the chance that the sample of / coins contains exactly r false ones?20. The Three-Cornered Duel A, B, and C are to fight a three-cornered pistol duel. All know that A's chance of hitting his target is 0.3, C's is 0.5, and B never misses. They are to fire at their choice of target in succession in the order A, B, C, cyclically (but a hit man loses further turns and is no longer shot at) until only one man is left unhit. What should A's strategy be?16. Will Second-Best Be Runner-Up? A tennis tournament has 8 players. The number a player draws from a hat decides his first-round rung in the tournament ladder See diagram. Suppose that the best player always defeats the next best and that the latter always defeats all the rest. The loser of the finals gets the runner-up cup. What is the chance that the second-best player wins the runner-up cup?13. The Prisoner's Dilemma 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 / 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 @. He thinks that if the warder says " will be released," his own chances have now gone down to , because either A and B or B and Care to be released. And so A decides not to reduce his chances by asking. However, A is mistaken in his calculations Explain
