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# Probability

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The Boomtown Bears are playing against the Tipton Toros in a baseball tournament. The winner of the tournament is the first team that wins three games. The Bears have a probability of of winning each game. Find the probability that the Bears win the tournament.

Mar 13, 2023

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To find the probability that the Bears win the tournament, we need to consider the different ways in which they can win. Since the first team to win three games wins the tournament, there are only two possible scenarios:

1. The Bears win the first three games.
2. The Bears win two games, and then win the third game.

The probability of the first scenario is (0.6)^3 = 0.216, since the probability of winning each game is 0.6 and we're assuming that the Bears win all three games in a row.

The probability of the second scenario can be found by considering the different orders in which the Bears can win two games, and then win the third game. The Bears can win the first two games and then the third game, or they can win the first and third games, or they can win the second and third games. The probability of each of these scenarios is:

- (0.6)^2 x 0.4 = 0.144 (Bears win first two games, then win third game)
- 0.6 x 0.4 x 0.6 = 0.144 (Bears win first and third games, then win second game)
- 0.4 x (0.6)^2 = 0.144 (Bears win second and third games, then win first game)

Therefore, the total probability of the second scenario is 0.144 + 0.144 + 0.144 = 0.432.

Adding the probabilities of the two scenarios, we get:

0.216 + 0.432 = 0.648

Therefore, the probability that the Bears win the tournament is 0.648, or 64.8%.

Mar 13, 2023