7. The Cubs play the Cardinals in the National League Championship Series in a b
ID: 3043900 • Letter: 7
Question
7. The Cubs play the Cardinals in the National League Championship Series in a best-of-seven series, i.e. a series which terminates when one of the two teams wins for the fourth time. Suppose that in each game played, the Cubs win with probability p independent of the outcomes of the other games (a) The Cubs win in 7 games if an only if they win exactly 3 of the first 6 games, and then win the seventh game. Use this reasoning to find, as a function of p, the probability that the Cubs win the series in 7 games. (b) Use the same reasoning to find, as a function of p, the probability that the Cubs win in 6 games; in 5 games; and in 4 games (c) Find the probability that the Cubs win the series in terms of p by summing the probabilities from (a) and (b) (d) Find the probability that the Cubs win any 4 of the 7 games in terms of p. That, is assume that the Cubs and the Cardinals play all 7 games, no matter what. How does this compare to your answer from (c)? (e) Returning to the best-of-7 scenario (that is, the series stops once a team has won 4 games) let random variable N be the number of games played in the series. Find the probability mass function for N (you will need to find the probability that the Cardinals win in n games!) (f) What happens to the pmf as p approaches 1? What happens to the pmf as p approaches 0?Explanation / Answer
(a)
Using binomial probability formula, the probability to win exactly 3 of first 6 games
= 6C3 * p3 * (1 - p)6-3
= 20 p3 (1 - p)3
The probability to win in 7th game is p
As, winning probability in a game is independent of the outcomes of other games, the probability that the Cub wins the series in 7 games is,
20 p3 (1 - p)3 * p = 20 p4 (1 - p)3
(b)
For 6 games,
Using binomial probability formula, the probability to win exactly 3 of first 5 games
= 5C3 * p3 * (1 - p)5-3
= 10 p3 (1 - p)2
The probability to win in 6th game is p
As, winning probability in a game is independent of the outcomes of other games, the probability that the Cub wins the series in 6 games is,
10 p3 (1 - p)2 * p = 10 p4 (1 - p)2
For 5 games,
Using binomial probability formula, the probability to win exactly 3 of first 4 games
= 4C3 * p3 * (1 - p)4-3
= 4 p3 (1 - p)
The probability to win in 5th game is p
As, winning probability in a game is independent of the outcomes of other games, the probability that the Cub wins the series in 5 games is,
4 p3 (1 - p)* p = 4 p4 (1 - p)
For 4 games,
Using binomial probability formula, the probability to win exactly 3 of first 3 games
= 3C3 * p3 * (1 - p)3-3
= p3
The probability to win in 4th game is p
As, winning probability in a game is independent of the outcomes of other games, the probability that the Cub wins the series in 4 games is,
p3* p = p4
(c)
Probability that Cub wins the series = Probability that Cub wins in 7 games + Probability that Cub wins in 6 games + Probability that Cub wins in 5 games + Probability that Cub wins in 4 games
= 20 p4 (1 - p)3 + 10 p4 (1 - p)2 + 4 p4 (1 - p) + p4
= p4 [ 20 (1 - p)3 + 10(1 - p)2 + 4 (1 - p) + 1]
= p4 [ 20 (1 - p + p2 + p3) + 10(1 - 2p + p2 ) + 4 (1 - p) + 1]
= p4 [35 - 44p + 30 p2+ 20p3]
(d)
Using binomial probability formula, the probability to win any of 4 of 7 games
= 7C4 * p4 * (1 - p)7-4
= 35 p4 * (1 - p)3
From (c), the probability to win any of 4 of 7 games is different than probability that Cub wins the series.
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.