Question

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?

Answer 1

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($A_1$) = 0.6

     Proportion of Policyholders having preferred policies, P($A_2$) = 0.3

     Proportion of Policyholders having ultra preferred policies, P($A_3$) = 0.1

Now, D = event of policyholder dying next year

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

      P(D/$A_1$) = 0.01.

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

 0.008.

Probability of policyholder dying given he had ultra preferred policies,

P(D/$A_3$) = 0.007.

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

The formula is given by, P($A_k$/D) = $\frac{P(A_k)P(D/A_k)}{\sum_{i=1}^{m} P(A_i)P(D/A_i)}$ ,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($A_1$/D)

   P($A_1$/D) = $\frac{P(A_1)P(D/A_1)}{ P(A_1)P(D/A_1)+P(A_2)P(D/A_2)+P(A_3)P(D/A_3)}$

                = $\frac{0.6\times 0.01}{0.6\times 0.01 + 0.3\times 0.008+0.1\times 0.007}$ = $\frac{0.006}{0.0091}$ = 0.6593 .

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

    P($A_2$/D) = $\frac{P(A_2)P(D/A_2)}{ P(A_1)P(D/A_1)+P(A_2)P(D/A_2)+P(A_3)P(D/A_3)}$

                 = $\frac{0.3\times 0.008}{0.6\times 0.01 + 0.3\times 0.008+0.1\times 0.007}$ = $\frac{0.0024}{0.0091}$ = 0.2637 .

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

     P($A_3$/D) = $\frac{P(A_3)P(D/A_3)}{ P(A_1)P(D/A_1)+P(A_2)P(D/A_2)+P(A_3)P(D/A_3)}$

                  = $\frac{0.1\times 0.007}{0.6\times 0.01 + 0.3\times 0.008+0.1\times 0.007}$ = $\frac{0.0007}{0.0091}$ = 0.0769 .