introduction to statistic chapter 9 page 266 question 40

Question

the world series ends whan a team wins 4 games. Suppose that sports analysis consider one team a bit stronger, with a 55% chance to win any individual game. Estimate the likelihood that the underdog wins the series.

Answer 1

We calculate the probability using tree method.

There 4 different branches.

 

1. The underdog wins 4 times - only one way this can be done. P( 4 wins) = .45^4

2. The underdog losses 1 game only.  To calculate the probability, we need to calculate in how many ways 1 loss can be done. Note, the last game is always win. So, we have 1 loss and there are 4 games (last one is win). We can do that loss in 4 different ways as shown below:

 

w w w l w

w w l w w

w l w w w

l w w w w

The other way to calculate is ₄C_{1 = 4!/((4-1)!•1!) =}

 

Therefore P ( 4 win, 1 loss) = 4 • .45⁴•.55

 

3. Two losses

   The number of ways the underdog will have two lost games is ₅C_{2 =5!/(5-2)!•2!) = 10}

               P( 4 win, 2 losses) = 10 • .45⁴ • .55²

 

4. Three losses

    The number of ways the underdog will have 3 lost games is ₆C₃ =6!/((6-3)!•3!) = 20.

               P( 4 wins, 3 losses) = 20 • .45⁴ • .55³

 

We add the probabilities in points 1-4 and get

P(underdog wins) = P( 4 wins) +P(4 wins, 1 loss) + P(4 wins, 2 losses) + P(4 wins, 3 losses) =.3917...