In a World Series, two teams play each other in at least four and at most seven games. The first team to win four games is the winner of the World Series. Assuming that both teams are equally matched, what is the probability that a World Series will be one (a) in four games? (b) in five games? (c) in six games? (d) in seven games? Explain.
To find the probability of a World Series ending in a specific number of games, we need to understand the possible outcomes and calculate the probability of each.
(a) To win the World Series in four games, one team needs to win the first four games. Since the World Series will be a minimum of four games, there is only one possible outcome that meets this condition. Therefore, the probability of a World Series ending in four games is 1 out of all possible outcomes.
(b) To win the World Series in five games, one team needs to win exactly four out of the first five games. There are five possible games that could be the deciding game, and any of the teams could win those games. Therefore, the probability of a World Series ending in five games is (5 choose 4) × 0.5^4 × 0.5^1, where "5 choose 4" represents the number of ways to choose four games out of five, and 0.5^4 and 0.5^1 represent the probabilities of each team winning and losing each game, respectively.
(c) To win the World Series in six games, one team needs to win exactly four out of the first six games. There are six possible games that could be the deciding game, and any of the teams could win those games. Therefore, the probability of a World Series ending in six games is (6 choose 4) × 0.5^4 × 0.5^2.
(d) To win the World Series in seven games, both teams should win three games each in the first six games, resulting in a tie. In the seventh game, the team that wins it will be the World Series winner. Therefore, the probability of a World Series ending in seven games is 0.5^6 × 0.5^1 × 0.5^1, where 0.5^6 represents the probability of both teams winning three games each in the first six games, and 0.5^1 × 0.5^1 represents the probability of each team winning or losing the seventh game.
Keep in mind that the assumption of both teams being equally matched simplifies the calculation, as it assumes that each team has a 50% chance of winning each game.
To calculate the probability of a World Series ending in a specific number of games, we need to consider the number of possible outcomes for each scenario.
(a) For the World Series to end in four games, one team must win the first four games. There are two possibilities for each game: Team A wins or Team B wins. Since the teams are equally matched, the probability of either team winning a particular game is 0.5 (or 1/2). So, the probability of the World Series ending in four games is (1/2) * (1/2) * (1/2) * (1/2) = 1/16.
(b) For the World Series to end in five games, one team must win the first three games, and the other team must win the next two games. Again, there are two possibilities for each game. So, the probability of the World Series ending in five games is (1/2) * (1/2) * (1/2) * (1/2) * (1/2) = 1/32.
(c) For the World Series to end in six games, one team must win the first three games, the other team must win the next two games, and then the first team must win the sixth game. The probability of this happening is (1/2) * (1/2) * (1/2) * (1/2) * (1/2) * (1/2) = 1/64.
(d) For the World Series to end in seven games, both teams must win an equal number of games in the first six (e.g., 3-3), and then one team must win the seventh game. The probability of this happening is (1/2) * (1/2) * (1/2) * (1/2) * (1/2) * (1/2) * (1/2) = 1/128.
In summary:
(a) The probability of the World Series ending in four games is 1/16.
(b) The probability of the World Series ending in five games is 1/32.
(c) The probability of the World Series ending in six games is 1/64.
(d) The probability of the World Series ending in seven games is 1/128.
These probabilities assume that both teams are equally matched and have an equal chance of winning each game.