The probability that a visit to a primary care physician’s (PCP) office results in neither lab work nor referral to a specialist is 35%. Of those coming to a PCP’s office, 30% are referred to specialists and 40% require lab work. Calculate the probability that a visit to a PCP’s office results in both lab work and referral to a specialist.

• 0.05
• 0.12
• 0.18
• 0.25
• 0.35

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

Solution: A

Let

[math]R[/math] = event of referral to a specialist

[math]L[/math] = event of lab work

We want to find

[[math]] \begin{align*} \operatorname{P}[R∩L] &= \operatorname{P}[R] + \operatorname{P}[L] – \operatorname{P}[R∪L] \\ &= \operatorname{P}[R] + \operatorname{P}[L] – 1 + \operatorname{P}[~(R∪L)] \\ &= \operatorname{P}[R] + \operatorname{P}[L] – 1 + \operatorname{P}[~R∩~L] \\ &= 0.30 + 0.40 – 1 + 0.35 \\ &= 0.05 . \end{align*} [[/math]]

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