Question

An insurance company divides its policy holders into three categories: low risk, moderate risk, and high risk. The low-risk policy holders account for 55% of the total number of people insured by the company. The moderate-risk policy holders account for 25%, and the high-risk policy holders account for 20%. The probabilities that a low-risk, moderate-risk, and high-risk policy holder will file a claim within a given year are respectively 5%, 15% and 60%. Given that a policy holder files a claim this year, what is the probability that the person is a moderate-risk policy holder?

Answer 1

0.2027 = 20.27% probability that the person is a moderate-risk policy holder

Explanation:

Conditional Probability

We use the conditional probability formula to solve this question. It is

`P(B|A) = (P(A \cap B))/(P(A))`

In which

P(B|A) is the probability of event B happening, given that A happened.

`P(A \cap B)` is the probability of both A and B happening.

P(A) is the probability of A happening.

In this question:

Event A: Policy holder files a claim.

Event B: The person is a moderate-risk policy holder.

Probability of filling a claim:

5% of 55%(low risk)

15% of 25%(moderate risk)

60% of 20%(high risk).

So

`P(A) = 0.05*0.55 + 0.15*0.25 + 0.6*0.2 = 0.185`

Files a claim and is moderate risk:

15% of 25%(moderate risk)

So

`P(A \cap B) = 0.15*0.25 = 0.0375`

Given that a policy holder files a claim this year, what is the probability that the person is a moderate-risk policy holder?

`P(B|A) = (P(A \cap B))/(P(A)) = (0.0375)/(0.185) = 0.2027`

0.2027 = 20.27% probability that the person is a moderate-risk policy holder