Question

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?

Answer 1

Let us define the following events:
S: A policyholder has the standard policy;

P: A policyholder has the preferred policy;

U: A policyholder has the ultra-preferred policy; and

D: A policyholder dies in the next year.

Now, we are given the following details:

"Of the company's policyholders, 30%are standard,50% are preferred, and 20% are ultra-preferred"

=> P(S) = 0.3 ; P(P) = 0.5 and P(U) = 0.2

"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"

=> P(D|S) = 0.015 ; P(D|P) = 0.002 and P(D|U) = 0.001

a)

The probability that a policyholder has the ultra-preferred policy and dies in the next year is given by:

P(U∩D) = P(U)*P(D|U) = 0.2*0.001 = 0.0002 [ANSWER]

b)

The probability that a policyholder dies in the next year is given by:

P(D) = P(S)*P(D|S) + P(P)*P(D|P) + P(U)*P(D|U)

= 0.3*0.015 + 0.5*0.002 + 0.2*0.001

= 0.0045 + 0.001 + 0.0002

= 0.0057 [ANSWER]

c)

The probability that the deceased policyholder was ultra-preferred given that the policyholder died is given by:

P(U|D) = P(U∩D)/P(D) = 0.0002/0.0057 = 2/57 = 0.03508772 [ANSWER]

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