An insurance company examines its pool of auto insurance customers and gathers the following information:

1. All customers insure at least one car.
2. 70% of the customers insure more than one car.
3. 20% of the customers insure a sports car.
4. Of those customers who insure more than one car, 15% insure a sports car.

Calculate the probability that a randomly selected customer insures exactly one car and that car is not a sports car.

• 0.13
• 0.21
• 0.24
• 0.25
• 0.30

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

Solution: B

Let M = event that customer insures more than one car and S = event that customer insurers a sports car. Then applying DeMorgan’s Law, compute the desired probability as:

[[math]] \begin{align*} P(M^c \cap S^c) = P[(M \cup S)^c] &= 1- P(M \cup S) = 1-[P((M) + P(S) - P(M \cap S)] \\ &= 1 - P(M) -P(S) + P(S | M) P(M) = 1-0.70-0.20 + (0.15)(0.70) \\ &= 0.205. \end{align*} [[/math]]

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