Question

An insurance company estimates their policyholders as follows:

• 30% of policyholders who only have a life insurance policy will renew next year.

• 50% of policyholders who only have a health insurance policy will renew next year.

• 70% of policyholders who have a life insurance and a health insurance policy will renew at least one of those policies next year.

The company’s data show that

• 40% of policyholders have a life insurance policy.

• 45% of policyholders have a health insurance policy.

• 25% of policyholders have a life insurance and a health insurance policy.

Using the company’s estimates, calculate the percentage of policyholders that will renew a life insurance or health insurance policy next year.

Answer 1

Solution :

Let probability of life insurance and Health insurance policy = P(L) and P(H)

and probability of renewal = P(R) from above P(L) = 0.40 , P(H) = 0.45 , P(L H) = 0.25

also P(R/L - H) = 0.30

P(R/H - L) = 0.50

P(R/L H) = 0.70

P(Having only life insurance policy ) = P(L) - P(L H) = P(L - H) = 0.40 - 0.25 = 0.15  

P(Having only health insurance policy)= P(H-L) = P(H) - P(L H)

= 0.45-0.25

= 0.20

Hence probability a policy holders will renew at least one policy next year = P(R)

P(R) = P(L-H) * P(R/L - H) + P(H-L) * P(R/H - L ) + P(L H) * P(R/L H)

= 0.15*0.30+0.20*0.50+0.70*0.25

= 0.045+0.1+0.175

= 0.32