An electronic system contains three cooling components that operate independently. The probability of each component’s failure is 0.05. The system will overheat if and only if at least two components fail.

Calculate the probability that the system will overheat.

• 0.007
• 0.045
• 0.098
• 0.135
• 0.143

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

Solution: A

The number of components that fail has a binomial(2, 0.05) distribution. Then,

[[math]] \operatorname{P}( N ≥ 2)= p(2) + p(3) = \binom{3}{2}(0.05)^2(0.95) + \binom{3}{3}(0.05)^3 = 0.00725. [[/math]]

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