Question

A hospital uses the Emergency Severity Index1 to classify its patients. This classification scheme has five categories from Level 1 (the patient requires resuscitation or other highly emergent care) to Level 5 (the patient requires non-urgent care).

At this hospital, 2% of patients are classified as Level 1, 7% are classified as Level 2, 30% are classified as Level 3,10% are classified as Level 5, and the remaining percentage of patients are classified as Level 4. At this hospital, 99% of Level 1 patients stay overnight, 90% of Level 2 patients stay overnight, 30% of Level 3 patients stay overnight, 10% of Level 4 patients stay overnight, and 1% of Level 5 patients stay overnight.

Kennedy (a nurse at this hospital) randomly selects a patient who is staying in the hospital overnight. What is the probability that this patient was initially classified as Level 1? Round to the three decimal places.

Hint: Construct a table with 10,000 hypothetical patients.

P(Level 1 GIVEN overnight) = ???

Answer 1

To find the probability that the patient was initially classified as Level 1 given they are staying overnight, we can use Bayes' theorem.

Let's construct a table with 10,000 hypothetical patients based on the given percentages:

Level | Classification Percentage | Overnight Percentage | Total Patients
---------------------------------------------------------------------
Level 1 | 2% | 99% | 200
Level 2 | 7% | 90% | 700
Level 3 | 30% | 30% | 3000
Level 4 | X% | 10% | X
Level 5 | 10% | 1% | 1000

To find the value of X (percentage of Level 4 patients), we can subtract the percentages of other levels from 100%:

X% = 100% - 2% - 7% - 30% - 10% = 51%

So, there are 5,100 Level 4 patients.

Now, let's calculate the total number of patients staying overnight:

Total overnight patients = (Level 1 overnight patients) + (Level 2 overnight patients) + (Level 3 overnight patients) + (Level 4 overnight patients) + (Level 5 overnight patients)
Total overnight patients = (200 * 99%) + (700 * 90%) + (3000 * 30%) + (5100 * 10%) + (1000 * 1%)
Total overnight patients = 19,800 + 63,000 + 9,000 + 510 + 10
Total overnight patients = 92,320

Now, let's calculate the probability that the patient was initially classified as Level 1 given they are staying overnight:

P(Level 1 GIVEN overnight) = (Number of Level 1 overnight patients) / (Total overnight patients)
P(Level 1 GIVEN overnight) = 200 / 92,320
P(Level 1 GIVEN overnight) ≈ 0.002164

Rounded to three decimal places, the probability that the patient was initially classified as Level 1 given they are staying overnight is approximately 0.002.