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) =
First, let's calculate the total number of patients staying overnight by summing the percentages of each level that stay overnight:
(2% of Level 1) + (90% of Level 2) + (30% of Level 3) + (10% of Level 4) + (1% of Level 5) = 2% + 90% + 30% + 10% + 1% = 133% (since 133% is the total, we can add in the remaining percentage of patients classified as Level 4 that stay overnight)
Next, we need to determine the percentage of overnight patients that are initially classified as Level 1. To do this, we divide the percentage of Level 1 patients by the total number of patients staying overnight:
(2% of Level 1) / (133% total overnight patients) = 2% / 133% = 0.015 (rounded to 3 decimal places)
Therefore, the probability that the patient Kennedy randomly selected was initially classified as Level 1 is 0.015.
To calculate 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 hypothetical patients:
Level 1: 2% of 10,000 patients = 200 patients (all stay overnight)
Level 2: 7% of 10,000 patients = 700 patients (90% stay overnight = 630 patients)
Level 3: 30% of 10,000 patients = 3,000 patients (30% stay overnight = 900 patients)
Level 4: Remaining percentage of patients (100% - (2% + 7% + 30% + 10%)) = 51% of 10,000 patients = 5,100 patients (10% stay overnight = 510 patients)
Level 5: 10% of 10,000 patients = 1,000 patients (1% stay overnight = 10 patients)
Total patients staying overnight = 200 + 630 + 900 + 510 + 10 = 2,250 patients
Now, we can calculate the probability of the patient being initially classified as Level 1 given they are staying overnight:
P(Level 1 GIVEN overnight) = P(Level 1 AND staying overnight) / P(staying overnight)
P(Level 1 AND staying overnight) = 200 / 10,000 = 0.02
P(staying overnight) = 2,250 / 10,000 = 0.225
P(Level 1 GIVEN overnight) = 0.02 / 0.225 ≈ 0.088
Therefore, the probability that the patient was initially classified as Level 1 given they are staying overnight is approximately 0.088 or 8.8% when rounded to three decimal places.
To find the probability that the patient was initially classified as Level 1 given that they are staying in the hospital overnight, we need to use Bayes' theorem:
P(Level 1 GIVEN overnight) = P(overnight GIVEN Level 1) * P(Level 1) / P(overnight)
Let's break down the components of this equation:
P(overnight GIVEN Level 1) = 99% = 0.99. This means that 99% of Level 1 patients stay overnight.
P(Level 1) = 2% = 0.02. This is the probability that a patient is initially classified as Level 1.
P(overnight) = P(overnight AND Level 1) + P(overnight AND Level 2) + P(overnight AND Level 3) + P(overnight AND Level 4) + P(overnight AND Level 5).
To find P(overnight AND Level i), we multiply the probability of being classified as Level i with the probability of staying overnight for that level. Then we sum these probabilities for all levels.
P(overnight AND Level 1) = P(Level 1) * P(overnight GIVEN Level 1) = 0.02 * 0.99 = 0.0198.
P(overnight AND Level 2) = P(Level 2) * P(overnight GIVEN Level 2) = 0.07 * 0.90 = 0.063.
P(overnight AND Level 3) = P(Level 3) * P(overnight GIVEN Level 3) = 0.30 * 0.30 = 0.09.
P(overnight AND Level 4) = P(Level 4) * P(overnight GIVEN Level 4) = P(Level 4) * 0.10.
P(overnight AND Level 5) = P(Level 5) * P(overnight GIVEN Level 5) = 0.10 * 0.01 = 0.001.
Now, we can calculate P(overnight):
P(overnight) = P(overnight AND Level 1) + P(overnight AND Level 2) + P(overnight AND Level 3) + P(overnight AND Level 4) + P(overnight AND Level 5)
= 0.0198 + 0.063 + 0.09 + P(Level 4) * 0.10 + 0.001.
Since the remaining percentage of patients is classified as Level 4, we can calculate it by subtracting the percentages of the other levels:
P(Level 4) = 1 - (P(Level 1) + P(Level 2) + P(Level 3) + P(Level 5)) = 1 - (0.02 + 0.07 + 0.30 + 0.10) = 0.51.
Now we can substitute the values into the equation:
P(Level 1 GIVEN overnight) = P(overnight GIVEN Level 1) * P(Level 1) / P(overnight)
= 0.0198 / (0.0198 + 0.063 + 0.09 + 0.51 * 0.10 + 0.001).
Calculating these values, we find:
P(Level 1 GIVEN overnight) = 0.0198 / (0.0198 + 0.063 + 0.09 + 0.051 + 0.001) ≈ 0.165.
Therefore, the probability that the patient was initially classified as Level 1 given that they are staying in the hospital overnight is approximately 0.165, rounded to three decimal places.