An insurance company issues life insurance policies in three separate categories: standard, preferred, and ultra - preferred. Of the company's policyholders, 30%are standard, 50% are preferred, and 20% are ultra-preferred. each standard policyholder has a probability 0.015 of dying in the next year, each preferred policyholder has probability 0.002 of dying in the next year, and each ultra-preferred policyholder has probability 0.001 of dying in the next year.

a) what is the probability that a policyholder has the ultra-preferred policy and dies in the next year?

b) what is the probability that a policyholder dies in the next year?

c) a policyholder dies in the next year. what is the probability that the deceased policyholder was ultra-preferred?

a) 0.0002

b) 0.0057

c) 0.0364

Step-by-step explanation:

Lets start by stating the probabilities of a person belonging to each policy:

Standard: 0.3

Preferred: 0.5

Ultra - Preferred: 0.2

The probability of person belonging to each policy AND dying in the next year:

Standard: 0.3 x 0.015 = 0.0045

Preferred: 0.5 x 0.002 = 0.001

Ultra - Preferred: 0.2 x 0.001 = 0.0002

a) The probability a ultra - preferred policy holder dies in the next year is 0.001. To find the probability of a person being both a ultra - preferred policy holder AND die in the next year is: 0.001 x 0.2 = 0.0002

b) The probability is given by adding the probabilities calculated before:

0.0045 + 0.001 + 0.0002 = 0.0057

c) We use the results above again. This is 0.0002 / (0.001 + 0.0045). The answer comes out to be 0.0364