A manufacturer produces computers and releases them in shipments of 100. From a shipment of 100, the probability that exactly three computers are defective is twice the probability that exactly two computers are defective. The events that different computers are defective are mutually independent.

Calculate the probability that a randomly selected computer is defective.

• 0.040
• 0.042
• 0.058
• 0.060
• 0.072

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

Solution: C

The number of defective computers has a binomial distribution with [math]n = 100[/math] and [math]p[/math] unknown. We have

[[math]] \begin{align*} \operatorname{P}(X = 3) = \operatorname{P}(X=2) \\ \binom{100}{3}p^3(1-p)^{97} = 2 \binom{100}{2}p^2(1-p)^{98} \\ 161700 p = 2(4950)(1 − p ) \\ 171600 p = 9900 \\ p = 9900/171600 = 0.058. \end{align*} [[/math]]

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