An insurance company issues life insurance policies in three separate categories: standard, preferred, and ultra-preferred. Of the company’s policyholders, 50% are standard, 40% are preferred, and 10% are ultra-preferred. Each standard policyholder has probability 0.010 of dying in the next year, each preferred policyholder has probability 0.005 of dying in the next year, and each ultra-preferred policyholder has probability 0.001 of dying in the next year. A policyholder dies in the next year.

Calculate the probability that the deceased policyholder was ultra-preferred.

• 0.0001
• 0.0010
• 0.0071
• 0.0141
• 0.2817

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: D

Let

[math]S[/math] = Event of a standard policy

[math]F[/math] = Event of a preferred policy

[math]U[/math] = Event of an ultra-preferred policy

[math]D [/math]= Event that a policyholder dies

Then

[[math]] \begin{align*} \operatorname{P}[U | D ] &= \frac{\operatorname{P}[ D | U ] \operatorname{P}[U ]}{\operatorname{P}[ D | S ] \operatorname{P}[ S ] + \operatorname{P}[ D | F ] \operatorname{P}[ F ] + \operatorname{P}[ D | U ] \operatorname{P}[U ]} \\ &= \frac{( 0.001)( 0.10 )}{( 0.01)( 0.50 ) + ( 0.005 )( 0.40 ) + ( 0.001)( 0.10 )} \\ &= 0.0141. \end{align*} [[/math]]

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.