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Homework, Notes 3 Section 4

Question 1

In honor of the baseball postseason: What is the probability a team wins a series?

The Philadelphia Phillies are playing in a five game series. Suppose that we believe the Phillies are the better team and that the games are iid. In particular, define

Xi = 1 if the Phillies win the ith game and 0 if they lose

and suppose that

Xi ~ Bernoulli(.6) iid

(a) Suppose that the teams play all five games. Then

Y = X1 + X2 + X3 + X4 + X5

is the number of games the Phillies win out of the five games played.

(Notice that each X=1 is a win and X=0 is a loss, so when we add them up we get the number of wins.)

According to the definition in the lecture notes, Y ~ Binomial(5,.6).

Open up a spreadsheet in Excel and in cells A2-A7 type the numbers 0, 1, 2, 3, 4, and 5. In cell B2, type “=BINOMDIST(A2,5,.6,0)” (without the quotes) and hit Enter. Then click on the lower right corner of the cell and drag downward to cell B7.

(This is exactly how I made the spreadsheet on slide 36 of the notes.)

The entries in cells are the probabilities of Philadelphia winning 0, 1, 2, 3, 4, and 5 games out of a possible 5 games played. In order to win the series, the Phillies must win 3 or more out of the 5 games. What is the probability the Phillies win the series?

(b) In postseason play, the teams do not always play all five games. If the Phillies win the first three games in the series, they win the series, and the last two games are not played. What is the probability the Phillies win the first three games?

(c) What is the probability that the Phillies wins the series in four games?

(Hint: For this to happen, the Phillies need to win three out of four games, but NOT the first three. There are THREE ways this can happen, so if you like you can calculate the probability of each and use “or means add”… but there may be an easier way!)

(d) What is the probability the Phillies win the series in five games?

(Hint: For this to happen, they must lose two of the first four games and win the fifth. There are SIX ways this can happen: the five games have to go WLWLW, LWLWW, WWLLW, etc… you could find the probability of all six and add them together, but once you do a couple of them I think you’ll see an easier way.)

(e) Using your answers to parts b—d, what is the probability the Phillies win the series? Does it matter whether all five games are played?

(f) Now suppose the Phillies were playing a seven game series and we still believe they are the better team (the games are still Bernoulli(.6) iid). Using Excel as you did in part (a), find the probability they win a seven game series.

If you are the better team, would you rather play a 5 game or a 7 game series? (Optional: Can you explain this intuitively?)

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