Question

An insurance company found that 45% of all insurance policies are terminated before their maturity date. Assume that 10 polices are randomly selected from the company’s policy database. Assume a Binomial experiment.

Required:

a) What is the probability that eight policies are terminated before maturity?

b) What is the probability that at least eight policies are terminated before maturity?

c) What is the probability that at most eight policies are not terminated before maturity?

Answer 1

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

Given $n=10, p=0.45, q=1-p=0.55$

a)

$$P(X=8)=\dbinom{10}{8}(0.45)^{8}(0.55)^{10-8}$$

$$=0.02288958944$$

b)

$$P(X\ge8)=P(X=8)+P(X=9)+P(X=10)$$

$$=\dbinom{10}{8}(0.45)^{8}(0.55)^{10-8}$$

$$+\dbinom{10}{9}(0.45)^{9}(0.55)^{10-9}$$

$$+\dbinom{10}{10}(0.45)^{10}(0.55)^{10-10}$$

$$=0.02739183926$$

c)

$$P(X\ge2)=1-P(X=0)-P(X=1)$$

$$=1-\dbinom{10}{0}(0.45)^{0}(0.55)^{10-0}$$

$$-\dbinom{10}{1}(0.45)^{1}(0.55)^{10-1}$$

$$=0.97674289875$$