Answer to Question #219512 in Statistics and Probability for mimi

Question

Assume that 20% of cars travelling on a motorway are grey in color

. (a) Find the probability that out of the next 10 cars passing a certain point on the motorway, at least 5 are grey. (8)

(b) Find the probability that for the following 5 cars to pass the point, none are grey. (6)

(c) If the colors of 25 cars passing on the motorway are recorded, what is the expected number and the variance of the number of grey cars?

Answer 1

Let $X=$ the number of grey cars: $X\sim Bin(n, p).$

(a) Given $n=10, p=0.2, q=1-p=0.8$

$$P(X\geq 5)=1-P(X=0)-P(X=1)$$

$$-P(X=2)-P(X=3)-P(X=4)$$

$$=1-\dbinom{10}{0}(0.2)^0(0.8)^{10-0}-\dbinom{10}{1}(0.2)^1(0.8)^{10-1}$$

$$-\dbinom{10}{2}(0.2)^2(0.8)^{10-2}-\dbinom{10}{3}(0.2)^3(0.8)^{10-3})$$

$$-\dbinom{10}{4}(0.2)^4(0.8)^{10-4}=0.0327934976$$

(b) Given $n=5, p=0.2, q=1-p=0.8$

$$P(X=0)=\dbinom{5}{0}(0.2)^0(0.8)^{5-0}=0.32768$$

(c) Given $n=25, p=0.2, q=1-p=0.8$

$$E[X]=np=25(0.2)=5$$

$$Var[X]=npq=25(0.2)(0.8)=4$$