An insurance company pays hospital claims. The number of claims that include emergency room or operating room charges is 85% of the total number of claims. The number of claims that do not include emergency room charges is 25% of the total number of claims. The occurrence of emergency room charges is independent of the occurrence of operating room charges on hospital claims.

Calculate the probability that a claim submitted to the insurance company includes operating room charges.

• 0.10
• 0.20
• 0.25
• 0.40
• 0.80

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

Solution: D

Let

[math]O[/math] = Event of operating room charges

[math]E[/math] = Event of emergency room charges

Then

[[math]] 0.85 = \operatorname{P}( O ∪ E ) = \operatorname{P}( O ) + \operatorname{P}( E ) − \operatorname{P}( O ∩ E ) = \operatorname{P}( O ) + \operatorname{P}( E ) − \operatorname{P}( O ) \operatorname{P}( E ). [[/math]]

Since [math]\operatorname{P}( E^c ) = 0.25 = 1 − \operatorname{P}( E )[/math] , it follows [math]\operatorname{P}( E ) = 0.75 . [/math]

So

[[math]] \begin{align*} 0.85 &= \operatorname{P}( O ) + 0.75 − \operatorname{P}( O )( 0.75 ) \\ \operatorname{P}( O )(1 − 0.75 ) &= 0.10 \\ \operatorname{P}( O ) &= 0.40 \end{align*} [[/math]]

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