Answer to Question #241897 in Statistics and Probability for Ravi

Question

The “Titans” cricket team has a winning rate of 75%. The team is planning to play 10 matches in the next season.

 

a)    Let X be the number of matches that will be won by the team. What are the possible values of X?

b)    What is the probability that the team will win exactly 6 matches?

c)    What is the probability that the team will lose 2 or less matches?

d)    What is the mean number of matches that the team will win?

e)    What are the variance and the standard deviation of the number of matches that the team will win?

Answer 1

a)

$$\{0,1,2,3,4,5,6,7,8,9,10\}$$

b) $X\sim Bin(n, p)$

$n=10, p=0.75, q=1-p=1-0.75=0.25$

$$P(X=6)=\dbinom{10}{6}(0.75)^6(0.25)^{10-6}$$

$$=0.1459980011$$

c)

$$P(X\geq8)=P(X=8)+P(X=9)$$

$$+P(X=10)=\dbinom{10}{8}(0.75)^8(0.25)^{10-8}$$

$$+\dbinom{10}{9}(0.75)^9(0.25)^{10-9}+\dbinom{10}{10}(0.75)^{10}(0.25)^{10-10}$$

$$=0.52559280396$$

d)

$$E(X)=mean=np=10(0.75)=7.5$$

(e)

$$Var(X)=\sigma^2=npq=10(0.75)(0.25)=1.875$$

$$\sigma=\sqrt{\sigma^2}=\sqrt{1.875}=1.3693$$