The number of injury claims per month is modeled by a random variable [math]N[/math] with

[[math]] \operatorname{P}[N=n] = \frac{1}{(n+1)(n+2)} [[/math]]

, for nonnegative integers, [math]n[/math]. Calculate the probability of at least one claim during a particular month, given that there have been at most four claims during that month.

• 1/3
• 2/5
• 1/2
• 3/5
• 5/6

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

Solution: B

Observe

[[math]] \begin{align*} \operatorname{P}[ N ≥ 1 | N ≤ 4 ] = \frac{\operatorname{P}[1 ≤ N ≤ 4]}{\operatorname{P}[ N ≤ 4]} &= \frac{1/6 + 1/12 + 1/20 + 1/30}{1/2 + 1/6 +1/12 + 1/20 + 1/30} \\ &= \frac{10 + 5 + 3 + 2}{30 + 10 + 5 + 3 + 2} \\ &= \frac{20}{50} \\ &= \frac{2}{5}. \end{align*} [[/math]]

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