Teams 1, 2, 3, 4 are all scheduled to play each of the other teams 10 times. Whenever team i plays team j, team i is the winner with probability P i , j where P 1 , 2 = .6 , P 1 , 3 = .7 , P 1 , 4 = .75 P 2 , 1 = .4 , P 2 , 3 = .6 , P 2 , 4 = .70 a. Approximate the probability that team 1 wins at least 20 games. Suppose we want to approximate the probability that team 2 wins at least as many games as does team 1. To do so, let X be the number of games that team 2 wins against team 1, let Y be the total number of games that team 2 wins against teams 3 and 4, and let Z be the total number of games that team 1 wins against teams 3 and 4. b. Are X, Y, Z independent. c. Express the event that team 2 wins at least as many games as does team 1 in terms of the random variables X,Y,Z. d. Approximate the probability that team 2 wins at least as many games as team 1. Hint: Approximate the distribution of any binomial random variable by a normal with the same mean and variance.
Teams 1, 2, 3, 4 are all scheduled to play each of the other teams 10 times. Whenever team i plays team j, team i is the winner with probability P i , j where P 1 , 2 = .6 , P 1 , 3 = .7 , P 1 , 4 = .75 P 2 , 1 = .4 , P 2 , 3 = .6 , P 2 , 4 = .70 a. Approximate the probability that team 1 wins at least 20 games. Suppose we want to approximate the probability that team 2 wins at least as many games as does team 1. To do so, let X be the number of games that team 2 wins against team 1, let Y be the total number of games that team 2 wins against teams 3 and 4, and let Z be the total number of games that team 1 wins against teams 3 and 4. b. Are X, Y, Z independent. c. Express the event that team 2 wins at least as many games as does team 1 in terms of the random variables X,Y,Z. d. Approximate the probability that team 2 wins at least as many games as team 1. Hint: Approximate the distribution of any binomial random variable by a normal with the same mean and variance.
Solution Summary: The author explains how to approximate the probability that team 1 wins at least 20 games.
Teams 1, 2, 3, 4 are all scheduled to play each of the other teams 10 times. Whenever team i plays team j, team i is the winner with probability
P
i
,
j
where
P
1
,
2
=
.6
,
P
1
,
3
=
.7
,
P
1
,
4
=
.75
P
2
,
1
=
.4
,
P
2
,
3
=
.6
,
P
2
,
4
=
.70
a. Approximate the probability that team 1 wins at least 20 games. Suppose we want to approximate the probability that team 2 wins at least as many games as does team 1. To do so, let X be the number of games that team 2 wins against team 1, let Y be the total number of games that team 2 wins against teams 3 and 4, and let Z be the total number of games that team
1 wins against teams 3 and 4.
b. Are X, Y, Z independent.
c. Express the event that team 2 wins at least as many games as does team 1 in terms of the random variables X,Y,Z.
d. Approximate the probability that team 2 wins at least as many games as team 1.
Hint: Approximate the distribution of any binomial random variable by a normal with the same mean and variance.
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, probability and related others by exploring similar questions and additional content below.
Discrete Distributions: Binomial, Poisson and Hypergeometric | Statistics for Data Science; Author: Dr. Bharatendra Rai;https://www.youtube.com/watch?v=lHhyy4JMigg;License: Standard Youtube License