Question

CASE-BASED/DATA-BASED

An insurance company believes that people can be divided into two classes: those who are accident prone and those who are not. The company’s statistics show that an accident-prone person will have an accident at some time within a fixed one-year period with a probability 0.6, whereas this probability is 0.2 for a person who is not accident prone. The company knows that 20 percent of the population is accident prone.

Based on the given information, answer the following questions.

1. What is the probability that a new policyholder will have an accident within a year of purchasing a policy?
2. Suppose that a new policyholder has an accident within a year of purchasing a policy. What is the probability that he or she is accident prone?

Answer 1

Let E₁ = The policyholder is accident prone.

E₂ = The policyholder is not accident prone.

E = The new policyholder has an accident within a year of purchasing a policy.

i. `"P"("E") = "P"("E"_1) × "P"("E"/"E"_1) + "P"("E"_2) × "P"("E"/"E"_2)`

= `20/100 xx 6/10 + 80/100 xx 2/10`

= `7/25`

ii. By Bayes’ Theorem, `"P"("E"_1/"E") = ("P"("E"_1) xx "P"("E"/"E"_1))/("P"("E"))`

= `(20/100 xx 6/10)/(280/1000)`

= `3/7`