An insurance company categorizes its policyholders into three mutually exclusive groups: high-risk, medium-risk, and low-risk. An internal study of the company showed that 45% of the policyholders are low-risk and 35% are medium-risk. The probability of death over the next year, given that a policyholder is high-risk is two times the probability of death of a medium-risk policyholder. The probability of death over the next year, given that a policyholder is medium-risk is three times the probability of death of a low-risk policyholder. The probability of death of a randomly selected policyholder, over the next year, is 0.009.

Calculate the probability of death of a policyholder over the next year, given that the policyholder is high-risk.

• 0.0025
• 0.0200
• 0.1215
• 0.2000
• 0.3750

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

Solution: B

[[math]] \begin{align*} \operatorname{P}[ D] &= \operatorname{P}[ H ]\operatorname{P}[ D | H ] + \operatorname{P}[ M ]\operatorname{P}[ D | M ] + \operatorname{P}[ L]\operatorname{P}[ D | L] \\ 0.009 &= \operatorname{P}[ H ]\operatorname{P}[ D | H ] + \operatorname{P}[M] \left( \frac{1}{2} \operatorname{P}[D | H] \right) + \operatorname{P}[L] \left( \frac{1}{2} \frac{1}{3} \operatorname{P}[D | H] \right) \\ 0.009 &= 0.20 \operatorname{P}[ D | H ] + 0.35 \left( \operatorname{P}[ D | H ] \right) + 0.45 \left( \operatorname{P}[ D | H ] \right) = 0.45 \operatorname{P}[ D | H ] \\ \operatorname{P}[ D | H ] &= 0.009/ 0.45 = 0.02 \end{align*} [[/math]]

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