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An archery game is played by shooting arrows at a target. Each person can choose to have two or three shots at the target. Oscar decides to play two games. In the first game he chooses to shoot two arrows and he wins if he hits the target at least once. In the second game, he chooses to shoot three arrows and wins if he hits the target at least twice. The probability that Oscar can hit the target on any shot is p, where 0 < p < 1. The probability that Oscar wins Game 1 is 2p - p² and the probability that he wins Game 2 is 3p²-2p³. Prove that Oscar is more likely to win Game 1 than Game 2 and find the exact value of p for which Oscar is twice as likely to win Game 1 than he is to win Game 2.

An archery game is played by shooting arrows at a target. Each person can choose to have two or three shots at the target. Oscar decides to play two games. In the first game he chooses to shoot two arrows and he wins if he hits the target at least once. In the second game, he chooses to shoot three arrows and wins if he hits the target at least twice. The probability that Oscar can hit the target on any shot is p, where 0 < p < 1. The probability that Oscar wins Game 1 is 2p - p² and the probability that he wins Game 2 is 3p²-2p³. Prove that Oscar is more likely to win Game 1 than Game 2 and find the exact value of p for which Oscar is twice as likely to win Game 1 than he is to win Game 2.

Algebra and Trigonometry (MindTap Course List)
Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter14: Counting And Probability
Section14.FOM: Focus On Modeling: The Monte Carlo Method
Problem 3P: Dividing a JackpotA game between two players consists of tossing a coin. Player A gets a point if...
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An archery game is played by shooting arrows at a target. Each person can choose to have two or three shots at the target. Oscar decides to play two games. In the first game he chooses to shoot two arrows and he wins if he hits the target at least once. In the second game, he chooses to shoot three arrows and wins if he hits the target at least twice. The probability that Oscar can hit the target on any shot is p, where 0 < p < 1. The probability that Oscar wins Game 1 is 2p - p² and the probability that he wins Game 2 is 3p²-2p³. Prove that Oscar is more likely to win Game 1 than Game 2 and find the exact value of p for which Oscar is twice as likely to win Game 1 than he is to win Game 2.
Transcribed Image Text:An archery game is played by shooting arrows at a target. Each person can choose to have two or three shots at the target. Oscar decides to play two games. In the first game he chooses to shoot two arrows and he wins if he hits the target at least once. In the second game, he chooses to shoot three arrows and wins if he hits the target at least twice. The probability that Oscar can hit the target on any shot is p, where 0 < p < 1. The probability that Oscar wins Game 1 is 2p - p² and the probability that he wins Game 2 is 3p²-2p³. Prove that Oscar is more likely to win Game 1 than Game 2 and find the exact value of p for which Oscar is twice as likely to win Game 1 than he is to win Game 2.
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