Question

An insurance company issues life insurance policies in three separate categories: standard, prefferred, and ultra-preferred. Of the company’s policyholders, 50% are standard, 40% are preffered, and 10% are ultra-prefferd. Each standard policyholder has probability 0.010 of dying in the next year, each preferred policyholder has probability 0.005 of dying in the next year, and each ultra-preferred policyholder has probability 0.001 of dying in the next year. Now, suppose a policyholder dies in the next year. What is the probability that the deceased policyholder was ultra-preferred

Answer 1

0.014

Explanation:

Hi!

Lets call:

D = {probability that a policyholder dies next year}

std = {policyholder is standard}

pref = {policyholder is preferred}

ultra = {policyholder is ultra-preferred}

We know that:

P(std) = 0.5

P(pref) = 0.4

P(ultra) = 0.1

P(D | std) = 0.01

P(D | std) = 0.005

P(D | std) = 0.001

We must find P(ultra | D). We can use Bayes theorem:

`P(ultra|D)=P(D|ultra)(P(ultra))/(P(D))\\P(D) = P(D|std)P(std) + P(D|pref)P(pref)+P(D|ultra)P(ultra) = 0.0071`

Then P(ultra | D) = 0.014