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) = ???
To solve this problem, we can use Bayes' theorem, which states:
P(A|B) = P(B|A) * P(A) / P(B)
Let's calculate the values needed to apply Bayes' theorem:
P(Level 1) = 2% = 0.02
P(overnight) = P(overnight given Level 1) * P(Level 1) + P(overnight given Level 2) * P(Level 2) + P(overnight given Level 3) * P(Level 3) + P(overnight given Level 4) * P(Level 4) + P(overnight given Level 5) * P(Level 5)
= 99% * 0.02 + 90% * 0.07 + 30% * 0.3 + 10% * (1 - 0.02 - 0.07 - 0.3 - 0.1) + 1% * 0.1
= 0.0198 + 0.063 + 0.09 + 0.006 + 0.001
= 0.1798
Now we can apply Bayes' theorem:
P(Level 1 given overnight) = P(overnight given Level 1) * P(Level 1) / P(overnight)
= 0.99 * 0.02 / 0.1798
≈ 0.010011
Rounding to three decimal places, the probability that the patient was initially classified as Level 1 given that they are staying overnight is approximately 0.010.
To find the probability that the patient was initially classified as Level 1 given that they are staying overnight, we can use Bayes' theorem, which states:
P(A|B) = (P(B|A) * P(A)) / P(B)
Here, A represents the event that the patient was initially classified as Level 1, and B represents the event that the patient is staying overnight.
Let's calculate each of the components in the formula:
P(A) = 2% = 0.02 (probability that a patient is initially classified as Level 1)
P(B|A) = 99% = 0.99 (probability that a Level 1 patient stays overnight)
P(B) = ? (probability that a patient stays overnight)
To calculate P(B), we need to consider the probability that a patient stays overnight for each level of severity, and then sum these probabilities:
P(B) = P(B|A) * P(A) + P(B|Level 2) * P(Level 2) + P(B|Level 3) * P(Level 3) + P(B|Level 4) * P(Level 4) + P(B|Level 5) * P(Level 5)
P(Level 2), P(Level 3), P(Level 4), and P(Level 5) represent the percentages given in the problem statement.
Now we can substitute the values into the formula:
P(Level 1|overnight) = (P(overnight|Level 1) * P(Level 1)) / P(overnight)
= (0.99 * 0.02) / P(B)
To find P(overnight), we can calculate it using the formula:
P(overnight) = P(overnight|Level 1) * P(Level 1) + P(overnight|Level 2) * P(Level 2) + P(overnight|Level 3) * P(Level 3) + P(overnight|Level 4) * P(Level 4) + P(overnight|Level 5) * P(Level 5)
P(Level 2), P(Level 3), P(Level 4), and P(Level 5) represent the percentages given in the problem statement.
Now we can substitute the values into the formula and solve for P(Level 1|overnight):
P(Level 1|overnight) = (0.99 * 0.02) / (P(overnight|Level 1) * P(Level 1) + P(overnight|Level 2) * P(Level 2) + P(overnight|Level 3) * P(Level 3) + P(overnight|Level 4) * P(Level 4) + P(overnight|Level 5) * P(Level 5))
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
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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.