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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.
+195 Since the winner of the tournament is the first team to win three games, there are five possible outcomes for the tournament:
1. The Bears win in three games.
2. The Toros win in three games.
3. The Bears win in four games.
4. The Toros win in four games.
5. The Bears win in five games.
To find the probability that the Bears win the tournament, we need to find the probability of each of these outcomes and add them up.
1. The probability that the Bears win in three games is (0.6)^3 = 0.216.
2. The probability that the Toros win in three games is (0.4)^3 = 0.064.
3. The probability that the Bears win in four games is the probability that they win three out of the first four games and lose the fifth game. This probability can be calculated as follows:
(3 choose 3) * (0.6)^3 * (0.4)^0 * (1 - 0.6) = 0.216 * 0.6 = 0.1296
4. The probability that the Toros win in four games can be calculated in the same way:
(3 choose 3) * (0.4)^3 * (0.6)^0 * (1 - 0.4) = 0.064 * 0.6 = 0.0384
5. The probability that the Bears win in five games is the probability that they win three out of the first four games and then win the fifth game. This probability can be calculated as follows:
(4 choose 3) * (0.6)^3 * (0.4)^1 * 0.6 = 0.144 * 0.4 = 0.0576
Adding up these probabilities, we get:
0.216 + 0.064 + 0.1296 + 0.0384 + 0.0576 = 0.5066
Therefore, the probability that the Bears win the tournament is approximately 0.5066, or 50.66%.
