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.

Step 1: Define Events and Probabilities

Let's define the following events for a patient's condition upon arrival and their outcome:

• C: The patient was categorized as Critical.
• S: The patient was categorized as Serious.
• T: The patient was categorized as Stable.
• D: The patient died.
• Surv: The patient survived.

We are given the following initial probabilities and conditional probabilities of death:

Initial Probabilities and Survival Rates

Category	Probability [math]P(\text{Category})[/math]	Probability of Death [math]P(D|\text{Category})[/math]	Probability of Survival [math]P(Surv|\text{Category}) = 1 - P(D|\text{Category})[/math]
Critical (C)	[math]P(C) = 0.10[/math]	[math]P(D|C) = 0.40[/math]	[math]P(Surv|C) = 1 - 0.40 = 0.60[/math]
Serious (S)	[math]P(S) = 0.30[/math]	[math]P(D|S) = 0.10[/math]	[math]P(Surv|S) = 1 - 0.10 = 0.90[/math]
Stable (T)	[math]P(T) = 1 - 0.10 - 0.30 = 0.60[/math]	[math]P(D|T) = 0.01[/math]	[math]P(Surv|T) = 1 - 0.01 = 0.99[/math]

Step 2: Calculate the Overall Probability of Survival

To use Bayes' Theorem, we first need to calculate the overall probability that a patient survived, denoted as [math]P(Surv)[/math]. We can achieve this by using the Law of Total Probability:

[[math]] P(Surv) = P(Surv|C)P(C) + P(Surv|S)P(S) + P(Surv|T)P(T) [[/math]]

Substituting the probabilities identified in Step 1:

[[math]] P(Surv) = (0.60)(0.10) + (0.90)(0.30) + (0.99)(0.60) [[/math]]

Performing the multiplication:

[[math]] P(Surv) = 0.06 + 0.27 + 0.594 [[/math]]

Summing these values gives the total probability of survival:

[[math]] P(Surv) = 0.924 [[/math]]

Step 3: Apply Bayes' Theorem

We want to calculate the probability that a patient was categorized as Serious, given that they survived, which is [math]P(S|Surv)[/math]. We can apply Bayes' Theorem using the probabilities calculated previously:

[[math]] P(S|Surv) = \frac{P(Surv|S)P(S)}{P(Surv)} [[/math]]

Substitute the values from Step 1 and Step 2 into the formula:

[[math]] P(S|Surv) = \frac{(0.90)(0.30)}{0.924} [[/math]]

Calculate the numerator:

[[math]] P(S|Surv) = \frac{0.27}{0.924} [[/math]]

Finally, perform the division:

[[math]] P(S|Surv) \approx 0.2922 [[/math]]

Rounding to two decimal places, the probability that the patient was categorized as serious upon arrival, given that they survived, is approximately [math]0.29[/math].

Key Insights

• Bayes' Theorem is crucial for calculating "reverse" conditional probabilities, i.e., [math]P(A|B)[/math] given [math]P(B|A)[/math] and prior probabilities.
• The Law of Total Probability is essential for finding the marginal probability of an event (e.g., [math]P(Surv)[/math]) by summing the probabilities of its occurrences across all mutually exclusive and exhaustive conditions.
• Careful definition of events and organization of given and derived probabilities (e.g., using a table) streamlines the problem-solving process and reduces errors.
• It's often more convenient to work with probabilities directly related to the question's outcome (e.g., survival probabilities) rather than their complements (death probabilities).

This article was generated by AI and may contain errors. If permitted, please edit the article to improve it.

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.