You own a light bulb factory. Your workforce is a bit clumsy – they keep dropping boxes of light bulbs. The boxes have varying numbers of light bulbs in them, and when dropped, the entire box is destroyed.

You are given:

• Expected number of boxes dropped per month: 50
• Variance of the number of boxes dropped per month: 100
• Expected value per box: 200
• Variance of the value per box: 400

You pay your employees a bonus if the value of light bulbs destroyed in a month is less than 8000.

Assuming independence and using the normal approximation, calculate the probability that you will pay your employees a bonus next month.

• 0.16
• 0.19
• 0.23
• 0.27
• 0.31

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

Key: A

[[math]] \begin{aligned} \operatorname{E}(S) &= \operatorname{E}(N)\operatorname{E}(X) = 50(200) = 10,000 \\ \operatorname{E}[ S ) &= \operatorname{E}[ N )\operatorname{E}[ X ) + \operatorname{E}[ X )^2\operatorname{E}[ N ) = 50(400) + 200 2 (100) = 4, 020, 000 \\ \operatorname{Pr}( S \lt 8, 000) &\approx \operatorname{Pr}( Z \lt \frac{8,000 - 10,000}{\sqrt{4,020,000}} = -0.998) = 0.16 \end{aligned} [[/math]]

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