From a Cramster Solution: Introductory Statistics (8th) ch. 5, prob 12E Question
ID: 2958527 • Letter: F
Question
From a Cramster Solution:Introductory Statistics (8th) ch. 5, prob 12E
Question Details
The World Series in baseball is won by the first team to win fourgames (ignoring the 1903 and 1919-1921 World Series, when it was abest of nine). Thus it takes at least four games and no morethan seven games to establish a winner. As found on the MajorLeague Baseball Web site in World Series Overview,historically, the lengths of the World Series are as given in thefollowing table.
Number ofGames Frequency Relative Frequency
4 17 0.175
5 23 0.237
6 22 0.227
7 35 0.361
a. If X denotes the number of games that it takes to completea World Series, identify the possible values of the random variableX.
b. Do the first and third columns of the table provide aprobability distribution for X? Explain your answer.
c. Historically, what is the most likely number of games ittakes to complete a series?
d. Historically, for a random chosen series, what is theprobability that it ends in five games?
e. Historically, for a randomly chosen series, what is theprobability 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 isahead three games to two, either team has the same chance ofwinning game number six. Thus there should be about an equalnumber of six and seven game series. If the teams are notevenly matched, the series should tend to be shorter, ending in sixor fewer games, not seven games. Can you explain why theseries tend to last longer than expected?
Explanation / Answer
a. X = 4, 5, 6, or 7 b. The first and third columns do define a probability distribution for X, since it associates each possible value of X with a probability of that outcome occurring. However, it does not provide a continuous probability function for X, since X can only take certain values. c. Historically, the series most often had 7 games (this has the highest probability, p = 0.361). d. The probability that a series ends in exactly 5 games is 0.237, or 23.7%. e. The probability that a series ends in 5 or more games is (0.237 + 0.227 + 0.361) = 0.825, or 82.5%. f. One reason that more series end in seven games than in six is that after 5 games, the series must be 3-2. That is, one team can win the series (and it ends in 6 games), or the other team can win and extend the series (and it ends in 7). The key is that the team who is could win the series potentially has less at stake - even if they lose, the series will continue. That is, the team in the lead effectively has two chances to win, while the team that is losing MUST win game six in order to have a chance at winning the series. Thus, while many athletes may disagree with this analysis, the losing team has a much stronger will to win game six than the winning team, and this apparently shows in the data - more often the trailing team wins!
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