At the start of a week, a coal mine has a high-capacity storage bin that is half full. During the week, 20 loads of coal are added to the storage bin. Each load of coal has a volume that is normally distributed with mean 1.50 cubic yards and standard deviation 0.25 cubic yards. During the same week, coal is removed from the storage bin and loaded into 4 railroad cars. The amount of coal loaded into each railroad car is normally distributed with mean 7.25 cubic yards and standard deviation 0.50 cubic yards. The amounts added to the storage bin or removed from the storage bin are mutually independent.

Calculate the probability that the storage bin contains more coal at the end of the week than it had at the beginning of the week.

• 0.56
• 0.63
• 0.67
• 0.75
• 0.98

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

Solution: D

With each load of coal having mean 1.5 and standard deviation 0.25, twenty loads have a mean of 20(1.5) = 30 and a variance of 20(0.0625) = 1.25. The total amount removed is normal with mean 4(7.25) = 29 and standard deviation 4(0.25) = 1. The difference is normal with mean 30 –29 = 1 and standard deviation sqrt(1.25 + 1) = 1.5. If D is that difference,

[[math]] \operatorname{P}( D \gt 0) = \operatorname{P}( Z \gt \frac{0-1}{1.5} = -0.67 ) = 0.7486. [[/math]]

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