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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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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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