1.5-6. A life insurance company issues standard, pre- ferred, and ultrapreferred

Question

1.5-6. A life insurance company issues standard, pre- ferred, and ultrapreferred policies. Of the company's policyholders of a certain age, 60% have standard poli- cies and 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 poli- cies 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?

Explanation / Answer

probability of dying in next year =P(standard policy and dies +preferred policy and dies +ultrapreferred policy and dies ) =0.60*0.01+0.3*0.008+0.1*0.007 =0.0091

hence probability of standard policy given dies=P(standard policy and dies)/P(dies)=0.6*0.01/0.0091=0.6593

probability of preferred policy given dies=P(preferred policy and dies)/P(dies)=0.3*0.008/0.0091=0.2637

probability of ultrapreferred policy given dies=P(ultrapreferred policy and dies)/P(dies)=0.1*0.007/0.0091=0.0769

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