Upon arrival at a hospital’s emergency room, patients are categorized according to their condition as critical, serious, or stable. In the past year:

1. 10% of the emergency room patients were critical;
2. 30% of the emergency room patients were serious;
3. the rest of the emergency room patients were stable;
4. 40% of the critical patients died;
5. 10% of the serious patients died; and
6. 1% of the stable patients died.

Given that a patient survived, calculate the probability that the patient was categorized as serious upon arrival.

• 0.06
• 0.29
• 0.30
• 0.39
• 0.64

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

Solution: B

Apply Baye’s Formula:

[[math]] \begin{align*} \operatorname{P}[Seri| Surv] &= \frac{\operatorname{P}[Surv. | Seri.] \operatorname{P}[Seri.]}{\operatorname{P}[Surv. | Crit.] \operatorname{P}[ Crit.] + \operatorname{P}[Surv. | Seri.] \operatorname{P}[Seri.] + \operatorname{P}[Surv. | Stab.] \operatorname{P}[Stab.]} \\ &= \frac{( 0.9 )( 0.3)}{( 0.6 )( 0.1) + ( 0.9 )( 0.3) + ( 0.99 )( 0.6 )} \\ &= 0.29. \end{align*} [[/math]]

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