A company establishes a fund of 120 from which it wants to pay an amount, C, to any of its 20 employees who achieve a high performance level during the coming year. Each employee has a 2% chance of achieving a high performance level during the coming year. The events of different employees achieving a high performance level during the coming year are mutually independent.

Calculate the maximum value of C for which the probability is less than 1% that the fund will be inadequate to cover all payments for high performance.

• 24
• 30
• 40
• 60
• 120

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

Solution: D

Let [math]X[/math] denote the number of employees who achieve the high performance level. Then [math]X[/math] follows a binomial distribution with parameters n = 20 and p = 0.02. Now we want to determine x such that [math]P[X \gt x] \lt 0.01 [/math] or equivalently

[[math]] 0.99 \leq P[X \leq x ] = \sum_{k=0}^{x} \binom{20}{k}(0.02)^k(0.98)^{20-k}. [[/math]]

The first three probabilities (at 0, 1, and 2) are 0.668, 0.272, and 0.053. The total is 0.993 and so the smallest [math]x[/math] that has the probability exceed 0.99 is 2. Thus [math]C = 120/2 = 60 [/math].

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