Toronto is playing Anaheim in the Stanley cup final. The first team to win four games is the new NHL champion. If the probability of that Toronto will win any game is 0.55 determine the probability that Anaheim wins the Stanley Cup.
To determine the probability that Anaheim wins the Stanley Cup, we need to consider various scenarios and calculate the probabilities accordingly.
First, let's consider how many games it will take for either team to win the championship. Since the first team to win four games becomes the champion, there are a few possible scenarios:
1. Toronto wins the series in exactly four games.
2. Toronto wins the series in exactly five games.
3. Toronto wins the series in exactly six games.
4. Toronto wins the series in exactly seven games.
5. Anaheim wins the series in exactly four games.
6. Anaheim wins the series in exactly five games.
7. Anaheim wins the series in exactly six games.
8. Anaheim wins the series in exactly seven games.
To calculate the probability of each scenario, we need to consider the probability of Toronto winning a specific number of games and the probability of Anaheim winning a specific number of games.
Given that the probability of Toronto winning any game is 0.55, we can calculate the probability of Anaheim winning any game as (1 - 0.55) = 0.45.
Now, let's calculate the probabilities for each scenario:
1. Probability of Toronto winning in four games: P(Toronto) * P(Toronto) * P(Toronto) * P(Toronto)
2. Probability of Toronto winning in five games: P(Toronto) * P(Toronto) * P(Toronto) * P(Anaheim) * P(Toronto)
3. Probability of Toronto winning in six games: P(Toronto) * P(Toronto) * P(Anaheim) * P(Toronto) * P(Anaheim) * P(Toronto)
4. Probability of Toronto winning in seven games: P(Toronto) * P(Anaheim) * P(Toronto) * P(Anaheim) * P(Toronto) * P(Anaheim) * P(Toronto)
5. Probability of Anaheim winning in four games: P(Anaheim) * P(Anaheim) * P(Anaheim) * P(Anaheim)
6. Probability of Anaheim winning in five games: P(Anaheim) * P(Anaheim) * P(Anaheim) * P(Toronto) * P(Anaheim)
7. Probability of Anaheim winning in six games: P(Anaheim) * P(Anaheim) * P(Toronto) * P(Anaheim) * P(Toronto) * P(Anaheim)
8. Probability of Anaheim winning in seven games: P(Anaheim) * P(Toronto) * P(Anaheim) * P(Toronto) * P(Anaheim) * P(Toronto) * P(Anaheim)
By substituting the probabilities, we can calculate the probability of each scenario. Finally, we add up the probabilities of scenarios 5, 6, 7, and 8 to get the probability that Anaheim wins the Stanley Cup.
Note: Calculating these probabilities depends on the assumption that each team's probability of winning a game remains constant throughout the series, which may not be entirely realistic.