In one company, 30% of males and 20% of females contribute to a supplemental retirement plan. Furthermore, 45% of the company’s employees are female.

Calculate the probability that a randomly selected employee is female, given that this employee contributes to a supplemental retirement plan.

• 0.09
• 0.23
• 0.35
• 0.45
• 0.55

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

Solution: C

Let C be the event that the employee contributes to a supplemental retirement plan and let F be the event that the employee is female. Then, by Bayes’ Theorem,

[[math]] \operatorname{P}(F | C) = \frac{\operatorname{P}(C | F ) \operatorname{P}( F )}{\operatorname{P}(C | F ) \operatorname{P}( F ) + \operatorname{P}(C | F ) \operatorname{P}( F )} = \frac{0.2(0.45)}{0.2(0.45) + 0.3(0.55)} = 0.353. [[/math]]

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