A study is being conducted in which the health of two independent groups of ten policyholders is being monitored over a one-year period of time. Individual participants in the study drop out before the end of the study with probability 0.2 (independently of the other participants).

Calculate the probability that at least nine participants complete the study in one of the two groups, but not in both groups.

• 0.096
• 0.192
• 0.235
• 0.376
• 0.469

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

Solution: E

Let

[math]X[/math] = number of group 1 participants that complete the study.

[math]Y[/math] = number of group 2 participants that complete the study.

Now we are given that [math]X[/math] and [math]Y[/math] are independent. Therefore,

[[math]] \begin{align*} \operatorname{P}[ [( X ≥ 9 ) ∩ ( Y \lt 9 )] \cup \operatorname{P}[( X \lt 9 ) ∩ ( Y ≥ 9 )] ] &= \operatorname{P}[ ( X ≥ 9 ) ∩ ( Y \lt 9 ) ] + \operatorname{P}[( X \lt 9 ) ∩ ( Y ≥ 9 ) ] \\ &= 2 \operatorname{P}[( X ≥ 9 ) ∩ ( Y \lt 9 ) ] \\ &= 2 \operatorname{P}[ X ≥ 9 ] \operatorname{P}[Y \lt 9 ] \\ &= 2 \operatorname{P}[ X ≥ 9] \operatorname{P}[ X \lt 9] \\ &= 2 \operatorname{P}[ X ≥ 9] (1 − \operatorname{P}[ X ≥ 9] ) \\ &= 2[\binom{10}{9}(0.2)(0.8)^9 + \binom{10}{10}(0.8)^{10}][1-\binom{10}{9}(0.2)(0.8)^9-\binom{10}{10}(0.8)^{10}]\\ &= 2 [ 0.376][1 − 0.376] = 0.469 \end{align*} [[/math]]

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