A machine has two parts labelled A and B. The probability that part A works for one year is 0.8 and the probability that part B works for one year is 0.6. The probability that at least one part works for one year is 0.9.

Calculate the probability that part B works for one year, given that part A works for one year.

• 1/2
• 3/5
• 5/8
• 3/4
• 5/6

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

Solution: C

Let A be the event that part A is working after one year and B be the event that part B is working after one year. Then,

[[math]] \operatorname{P}(B | A) = \frac{\operatorname{P}(A \cap B)}{\operatorname{P}(A)} = \frac{\operatorname{P}( A) + \operatorname{P}( B ) − \operatorname{P}( A \cup B )}{\operatorname{P}(A)} = \frac{0.8 + 0.6 − 0.9}{0.8} = 5/8. [[/math]]

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