Problem:
Two teams are in a best-two-out-of-three playoff: the teams will play at most 3 games, and the winner of the playoff is the first team to win 2 games. The first game is played on Team A's home field, and the remaining games are played on Team B's home field. Team A has a chance of winning at home, and its probability of winning when playing away from home is . Outcomes of the games are independent. The probability that Team A wins the playoff is . Then can be written in the form , where and are positive integers. What is ?
Answer Choices:
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Solution:
There are three ways for Team A to win the playoff: win the first two games; win the first game, lose the second game, and win the third game; or lose the first game and win the second and third games. The probability that it wins in one of these ways is
Setting this equal to and simplifying gives , and the Quadratic Formula gives solutions . Choosing the plus sign gives a nonsensical value of because it is greater than 1 , so the required probability is . The requested sum is .
The problems on this page are the property of the MAA's American Mathematics Competitions