Stat 104 Section 4 - Problem Solutions

Stat 104 Section 4 - Problem Solutions - Stat 104 Section...

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Stat 104 Section Notes 4 Problem Solutions 1) In a 7 game playoff series like the World Series, the first team to win 4 games wins the series. a) Assuming there is equal probability that each team wins each game, what is the probability that the series ends in 4, 5, 6, or 7 games? Let X = number of games the World Series will go this year. Let there be two teams in the series: the Phillies (P) and the Red Sox (R). Then, for the series to end in four games with the Phillies as champions, the only way this can happen is if the Phillies win the first four games, so the sequence of game-by-game winners would be: P P P P . In each of those four games, the Phillies have 0.5 chance of winning each game, so the probability that the Phillies win in 4 games is (0.5) 4 . The same argument could be made for the Red Sox winning in 4 games (and thus we just multiply the (0.5) 4 times two): . For the series to end in five games with the Phillies as champions, there are 4 ways (game-by-game sequences) this can happen: 1) R P P P P , 2) P R P P P , 3) P P R P P , 4) P P P R P [Note, the Phillies have to win the 5 th game]. This is equivalent to choosing which 3 of the first 4 games the Phillies must win, thus there are ways to do this. Also in each of those five games, the Phillies have 0.5 chance of winning or losing each game, so the probability that the Phillies win in 5 games is 4*(0.5) 4 . The same argument could be made for the Red Sox winning in 5 games (and thus we just multiply the 4*(0.5) 4 times two): . Similar arguments can be made to calculate the remaining two possibilities as such: or Note: coming up with these calculations is difficult, and would be too challenging for a midterm question (though you would be expected to do them in a HW problem since you would have more time to think). b) Let X be the random variable for the number of games the World Series will go this year. What is the mean and variance of X ? Since X is a discrete random variable, we can use the following formulas: Mean:
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Variance: Thus, the standard deviation is: Since the World Series went to 7 games (in 1923), the number of games played has been: 4 games 5 games 6 games 7 games Frequenc y 17 18 18 33 c) Anything interesting seen above?
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Stat 104 Section 4 - Problem Solutions - Stat 104 Section...

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