Question

An insurance company has three types of annuity products: indexed annuity, fixed annuity, and variable annuity. You are given:

• None of the customers have both fixed annuity and variable annuity.
• 40% of the customers with fixed annuity also have indexed annuity.
• Half of the customers with two annuity products have variable annuity.
• 60% of the customers have indexed annuity.
• The number of customers who have only the variable annuity is the same as the number of customers who have two annuity products.
• All customers have at least one type of annuity.

Determine the proportion of the customers who only have the indexed annuity.

Answer 1

Let I, F and V denote the event of a customer having Indexed annuity, fixed annuity and variable annuity respectively

P(F and V) = 0

Thus, P(V and F and I) = 0

P(I | F) = 0.4

-> P(I and F)/P(F) = 0.4

-> P(F) = 2.5P(I and F)

P(V and I and not F) = 0.5{P(V and I and not F) + P(I and F and not V) + P(F and V and not I)}

-> P(V and I and not F) = P(I and F and not V)

-> P(V and I) = P(I and F) (since P(V and F and I) = 0)

P(I) = 0.6

P(V only) = P(V and I) + P(I and F)

= 2P(I and F)

P(V) = P(V only) + P(V and I) + P(V and F) - P(V and F and I)

= 2P(I and F) + P(I and F) = 3P(I and F)

P(V or I or F) = 1

-> P(V) + P(I) + P(F) - P(V and I) - P(I and F) - P(F and V) + P(F and V and I) = 1

-> 3P(I and F) + 0.6 + 2.5P(I and F) - P(I and F) - P(I and F) = 1

-> 3.5P(I and F) = 0.4

-> P(I and F) = 4/35

P(I only) = P(I) - P(V and I) - P(I and F)

= P(I) - 2P(I and F)

The proportion of the customers who only have the indexed annuity = 13/35