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Statistics / Introduction to Probability, 2 / Chapter 1 / Problem 8

Introduction to Probability, | 2nd Edition | ISBN: 9781886529236 | Authors: Dimitri P. Bertsekas John N. Tsitsiklis

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You enter a special kind of chess tournament, in which you

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QUESTION:

You enter a special kind of chess tournament, in which you play one game with each of three opponents, but you get to choose the order in which you play your opponents, knowing the probability of a win against each. You win the tournament if you win two games in a row, and you want to maximize the probability of winning. Show that it is optimal to play the weakest opponent second, and that the order of playing the other two opponents does not matter.

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QUESTION:

You enter a special kind of chess tournament, in which you play one game with each of three opponents, but you get to choose the order in which you play your opponents, knowing the probability of a win against each. You win the tournament if you win two games in a row, and you want to maximize the probability of winning. Show that it is optimal to play the weakest opponent second, and that the order of playing the other two opponents does not matter.

ANSWER:

Step 1 of 2

We assume the plays with different players are independent. Define and (i=1,2,3)

as winning and losing over the players respectively, and let pi denote the probability of winning over the player, then

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