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?

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() = 0.6

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

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

Now, D = event of policyholder dying next year

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

      P(D/) = 0.01.

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

 0.008.

Probability of policyholder dying given he had ultra preferred policies,

P(D/) = 0.007.

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

The formula is given by, P(/D) = ,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(/D)

   P(/D) =

                = = = 0.6593 .

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

    P(/D) =

                 = = = 0.2637 .

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

     P(/D) =

                  = = = 0.0769 .

d. it is a convex pentagon because it has five sides and none of the sides would extend into the inside of the polygon.

step-by-step explanation: