A theme park conducts a study of families that visit the park during a year. The fraction of such families of size [math]m[/math] is [math]\frac{8-m}{28}, \, m = 1,\ldots,7[/math]

For a family of size [math]m[/math] that visits the park, the number of members of the family that ride the roller coaster follows a discrete uniform distribution on the set [math]\{1,\ldots, m\}[/math].

Calculate the probability that a family visiting the park has exactly six members, given that exactly five members of the family ride the roller coaster.

• 0.17
• 0.21
• 0.24
• 0.28
• 0.31

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

Solution: E

Let M be the size of a family that visits the park and let N be the number of members of that family that ride the roller coaster. We want [math]\operatorname{P}(M = 6 | N = 5).[/math] By Bayes theorem

[[math]] \begin{align*} \operatorname{P}(M = 6 | N = 5) &= \frac{\operatorname{P}(N = 5 | M = 6) \operatorname{P}(M =6)}{\sum_{m=1}^7 \operatorname{P}(N =5 | M=m) \operatorname{P}(M=m)} \\ &= \frac{\frac{1}{6} \frac{2}{28}}{0 + 0 + 0 + 0 + \frac{1}{5} \frac{3}{28} + \frac{1}{6}\frac{3}{28} + \frac{1}{6}\frac{2}{28} + \frac{1}{7}\frac{1}{28}} \\ &= \frac{\frac{1}{3}}{\frac{3}{5} + \frac{1}{3} + \frac{1}{7}} = \frac{35}{63 + 35 + 15} = \frac{35}{113} \approx 0.3097. \end{align*} [[/math]]

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