Question

A research was conducted in 2017 at the Tema Port to find the ages of cars imported into

the country. According to the research, 10% of the cars imported were less than one-year-

old. Assuming this result holds true for the current period for all cars imported into the

country.

Find the probability that in a random sample of 5 cars at the Tema Port

(i) exactly 3 are less than one-year-old.

(ii) exactly 2 are less than one-year-old.

(iii) none is less than one-year-old

(iv) Find the mean and standard deviation of the distribution if a random

Sample of 200 cars were selected at Tema Port

Answer 1

We have that:

p = 10% = 0.1

n = 5

This follows the binomial distribution.

The binomial probability is calculated by the formula:

$$P(X=m)=C(n,m)\cdot p^m \cdot(1-p)^{n-m}$$

i) m = 3

$P(X=3) = C(5,3)\cdot 0.1^3 \cdot 0.9^{5-3}=\frac{5!}{3!2!}\cdot0.1^3\cdot0.9^2=0.0081$

ii) m = 2

$P(X=2) = C(5,2)\cdot 0.1^2 \cdot 0.9^3=0.0729$

iii) m = 0

$P(X=0) = C(5,9)\cdot 0.1^0 \cdot 0.9^5=0.59049$

iv) n = 200

The mean is calculated by the formula:

$\mu = np=200\cdot 0.1=20$

The standard deviation is calculated by the formula:

$\sigma=\sqrt{np(1-p)}=\sqrt{200\cdot0.1\cdot0.9}=4.24$