Q 5.118 World Series. The World Series i... [FREE SOLUTION]

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Chapter 5: Q 5.118 (page 225)

World Series. The World Series in baseball is won by the first team to win four games (ignoring the 1903 and 1919鈥�1921 World Series, when it was a best of nine). Thus it takes at least four games and no more than seven games to establish a winner. From the document World Series History on the Baseball Almanac website, as of November 2013, the lengths of the World Series are as given in the following table
Number of Games Frequency Relative Frequency
4 21 0.200
5 24 0.229
6 24 0.229
7 36 0.343
a. If X denotes the number of games that it takes to complete a World Series, identify the possible values of the random variable X.
b. Do the first and third columns of the table provide a probability distribution for X? Explain your answer.
c. Historically, what is the most likely number of games it takes to complete a series?
d. Historically, for a randomly chosen series, what is the probability that it ends in five games?
e. Historically, for a randomly chosen series, what is the probability that it ends in five or more games?
f. The data in the table exhibit a statistical oddity. If the two teams in a series are evenly matched and one team is ahead three games to two, either team has the same chance of winning game number six. Thus there should be about an equal number of six-and seven-game series. If the teams are not evenly matched, the series should tend to be shorter, ending in six or fewer games, not seven games. Can you explain why the series tend to last longer than expected?

Short Answer

Expert verified

Part a. X = {4,5,6,7}

Part b. Yes

Part c. 7 number of games

Part d. P(X=5) = 0.229

Part e. P(X 鈮� 5) = 0.801

Part f. Perhaps as a result of the manager's approach.

Step by step solution

Part (a) Step 1. Given information

The number of games is given.

Number of Games	Frequency	Relative Frequency
4	21	0.200
5	24	0.229
6	24	0.229
7	36	0.343

Part (a) Step 2. Solution

The number of matches is represented by X.

As stated in the question, the series could be completed by four, five, six, or seven matches. As a result, there is no limit to the number of matches that can be played.

So, the X will be {4,5,6,7}

Part (b) Step 1. Solution

Yes, the probability distribution is provided by the first and third columns of the table, because the specified relative frequency equals the likelihood of the games.

Part (c) Step 1. Solution

Which number of games will be played at the highest frequency and relative frequency. As a result, 7 games have the highest frequency and relative frequency, respectively 36 and 0.343.

Part (d) Step 1. Solution

The probability is calculated using the relative frequency of five games.:

P(X = 5) = 0.229

Part (e) Step 1. Solution

Using the addition rule for 5 or more games in a row.

P(A or B) = P(A) + P(B)

Then,

P(X鈮� 5 ) = P(X=5) + P(X=6) +P(X=7) = 0.229 + 0.229 +0.343

After simplifying we will get

P(X鈮� 5 ) = 0.801

Part (f) Step 1. Solution

This could occur as a result of the manager's approach. There's a probability that the manager who's ahead of schedule will decide to give the best pitch one more day of rest by omitting games 6 and 7.