In a shipment of 20 packages, 7 packages are damaged. The packages are randomly inspected, one at a time, without replacement, until the fourth damaged package is discovered.

Calculate the probability that exactly 12 packages are inspected.

• 0.079
• 0.119
• 0.237
• 0.243
• 0.358

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

Solution: B

The requested probability can be determined as P(3 of first 11 damaged) P(12^th is damaged | 3 of first 11 damaged)

[[math]] = \frac{\binom{7}{3} \binom{13}{8}}{\frac{20}{11}} \frac{4}{9} = \frac{351(1287)}{167960} \frac{4}{9} = 0.119. [[/math]]

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