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s49 - Solution to Problem 49 There were very many...

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Solution to Problem 49 There were very many submissions to this week’s problem. The only correct solution was sent in by Robert T. McQuaid. Way to go Robert! The incorrect submissions all made the assumption -- incorrect, it turns out -- that each man’s best strategy is to shoot at the other man with the better aim. However, as R. McQuaid correctly observes, this is not the best strategy for Sir Soso; in fact, if after drawing lots, Soso is to fire first, his best strategy is to miss on purpose! In every case, it is in Señor Casisiempre’s best interest to fire at Monsieur Toujours, should Toujours still be a threat; similarly it is in Toujours best interest to shoot at Casisiempre. A straightforward check of probabilities gives the following table. Firing Order Survival Probability for Toujours Casisiempre Soso T - C - S 1/2 0 1/2 T - S - C 1/2 0 1/2 C - T - S 1/10 16/45 49/90 C - S - T 1/10 16/45 49/90 S - T - C 1/2 0 1/2 S - C - T 1/10 16/45 49/90 In order to compute these probabilities, you might start out by figuring the likelihood of winning each possible two-man duel. The only tricky duel, and the only one that
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