An insurance company issues life insurance policies in three separate categories

Question

An insurance company issues life insurance policies in three separate categories: standard, preferred, and ultra-preferred. Of the company's policyholders, 50% are standard, 40% are preferred, and 10% are ultra-preferred. Each standard policyholder has probability 0.010 of dying this year, each preferred policyholder has probability 0.005 of dying this year and each ultra-preferred policyholder has probability 0.001 of dying this year. Given a policyholder died this year What is the probability that the deceased policyholder was ultra-preferred?

Explanation / Answer

Let

S = standard
P = preferre
U = ultra-preferred
D = dying

Thus,

P(U|D) = P(U n D) / P(D)

As

P(D) = P(S) P(D|S) + P(P) P(D|P) + P(U) P(D|U) = 0.50*0.010 + 0.40*0.005 + 0.10*0.001 = 0.0071

and

P(U n D) = P(U) P(D|U) = 0.10*0.001 = 0.0001

Thus,

P(U|D) = 0.0001/0.0071 = 0.014084507 [answer]

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