A life insurance company issues standard, preferred, and ultrapreferred policies. Of the company’s policyholders of a certain age, 60and a probability of 0.01 of dying in the next year, 30% have preferred policies and a probability of 0.008 of dying in the next year, and 10% have ultrapreferred policies and a probability of 0.007 of dying in the next year. A policyholder of that age dies in the next year. What are the conditional probabilities of the deceased having had a standard, a preferred, and an ultrapreferred policy?

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Alcaa Alcaa · Answer 1 · 2020-02-17T06:22:05+01:00

Answer:

a) The conditional probability of the deceased having had a standard policy = 0.6593

b) The conditional probability of the deceased having had a preferred policy = 0.2637

c) The conditional probability of the deceased having had an ultra preferred policy = 0.0769

Step-by-step explanation:

We are given that a life insurance company issues standard, preferred, and ultra preferred policies to it's policyholders.

Let Proportion of Policyholders having standard policies, P([tex]A_1[/tex]) = 0.6

Proportion of Policyholders having preferred policies, P([tex]A_2[/tex]) = 0.3

Proportion of Policyholders having ultra preferred policies, P([tex]A_3[/tex]) = 0.1

Now, D = event of policyholder dying next year

So, Probability of policyholder dying given he had standard policies,

P(D/[tex]A_1[/tex]) = 0.01.

Probability of policyholder dying given he had preferred policies, P(D/[tex]A_2[/tex]) =

0.008.

Probability of policyholder dying given he had ultra preferred policies,

P(D/[tex]A_3[/tex]) = 0.007.

Now using Bayes' Theorem we will find the required conditional probability;

The formula is given by, P([tex]A_k[/tex]/D) = [tex]\frac{P(A_k)P(D/A_k)}{\sum_{i=1}^{m} P(A_i)P(D/A_i)}[/tex] ,where i goes from 1 to 3.

a) Probability of the deceased having had a standard policy given he died in the next year = P([tex]A_1[/tex]/D)

P([tex]A_1[/tex]/D) = [tex]\frac{P(A_1)P(D/A_1)}{ P(A_1)P(D/A_1)+P(A_2)P(D/A_2)+P(A_3)P(D/A_3)}[/tex]

= [tex]\frac{0.6\times 0.01}{0.6\times 0.01 + 0.3\times 0.008+0.1\times 0.007}[/tex] = [tex]\frac{0.006}{0.0091}[/tex] = 0.6593 .

b) Probability of the deceased having had a preferred policy given he died in the next year = P([tex]A_2[/tex]/D)

P([tex]A_2[/tex]/D) = [tex]\frac{P(A_2)P(D/A_2)}{ P(A_1)P(D/A_1)+P(A_2)P(D/A_2)+P(A_3)P(D/A_3)}[/tex]

= [tex]\frac{0.3\times 0.008}{0.6\times 0.01 + 0.3\times 0.008+0.1\times 0.007}[/tex] = [tex]\frac{0.0024}{0.0091}[/tex] = 0.2637 .

c) Probability of the deceased having had a ultra preferred policy given he died in the next year = P([tex]A_3[/tex]/D)

P([tex]A_3[/tex]/D) = [tex]\frac{P(A_3)P(D/A_3)}{ P(A_1)P(D/A_1)+P(A_2)P(D/A_2)+P(A_3)P(D/A_3)}[/tex]

= [tex]\frac{0.1\times 0.007}{0.6\times 0.01 + 0.3\times 0.008+0.1\times 0.007}[/tex] = [tex]\frac{0.0007}{0.0091}[/tex] = 0.0769 .