Your favorite team is in the final playoffs You have assigne

Your favorite team is in the final playoffs. You have assigned a probability of 60% that it will win the championship. Past records indicate that when teams win the championship, they win the first game of the series 70% of the time. When they lose the series, they win the first game 25% of the time. The first game is over; your team has lost. What is the probability that it will win the series?

Solution

First we will define events. Let W be the event that your team wins the championship and F be the event that they win the first game (so Wc is the event they lose the championship and Fc is the event that they lose the first game). We are given: P(W) = 0.6 , P(F|W) = 0.7 , P(F|W\') = 0.25 Now, since we want to compute P(W|F\') =P(W \\ F\')(F\'),we only need to find P(W \\F\')and P(F\'). P(W \\ F\') = P(F\'|W) × P(W) = (0.3) × (0.6) = 0.18 We note that P(F\') = P(W \\ F\') + P(Wc \\ F\') (the team either loses the first game and wins the series or loses the first game and loses the series, and these are mutually exclusive events). We already know P(W \\ F\'), so we only need to find P(W\'capF\'). P(W\' \\ F\') = P(F\'|W\') × P(W\') = (0.75) × (0.4) = 0.3 So P(F\') = 0.18 + 0.3 = 0.48. Thus, P(W|F\') =P(W \\ F\')*P(F\')=0.18/0.48= 0.375.