statistics question

Question

Your favorite team is in the World Series. You have assigned a probability of 61% that they will win the championship. Past records indicate that when teams win the championship, they win the first game of the series 72% of the time. When they lose the championship, they win the first game 25% of the time. The first game is over and your team has lost. What is the probability that they will win the World Series?

Answer 1

This is a Bayes Theorem question.  First listing the given events, we have:

• Pr(Win the championship) = .61
• Pr(Win the first game | Win the championship) = .72
• Pr(Win the first game | Lose the championship) = .25

 

From these we also know/can derive:

• Pr(Lose the championship) = 1-.61 = .39
• Pr(Lose the first game | Win the championship) = 1-.72 = .28
• Pr(Lose the first game | Lose the championship) = 1-.25 = .75

 

And we want Pr(win the championship | Lose the first game) = ?

 

Using Bayes theorem, Pr(Win the championship | Lose the first game) = [Pr(Lose the first game | Win the championship) × Pr(Win the championship)] ÷ [Pr(Lose the first game | Win the championship) × Pr(Win the championship) + Pr(Lose the first game | Lose the championship) × Pr (Lose the championship)]

 

Therefore we have Pr (Win the championship | Lose the first game) = (.28)(.61) ÷ [(.28)(.61) + (.75)(.39)] = .3686596

 

You could have also used a tree diagram to help reach the answer as some people find it more intuitive then.  However, with enough practice and confidence, skipping the tree diagram can save you time on problems that would have more that just four possible event outcomes like this one had.