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.
Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.
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