A delivery service owns two cars that consume 15 and 30 miles per gallon. Fuel costs 3 per gallon. On any given business day, each car travels a number of miles that is independent of the other and is normally distributed with mean 25 miles and standard deviation 3 miles.

Calculate the probability that on any given business day, the total fuel cost to the delivery service will be less than 7.

• 0.13
• 0.23
• 0.29
• 0.38
• 0.47

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

Solution: B

Let X and Y be the miles driven by the two cars. The total cost, is then C = 3(X/15 + Y/30) = 0.2X + 0.1Y. C has a normal distribution with mean 0.2(25) + 0.1(25) = 7.5 and variance 0.04(9) + 0.01(9) = 0.45. Then

[[math]] \operatorname{P}(C \lt 7) = \operatorname{P}(Z \lt (7-7.5)/\sqrt{0.45}) = -0.7454) = 0.23. [[/math]]

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