In a clinic, 70% of patients are vaccinated against the flu. Among vaccinated patients, 90% do not contract the flu, while among unvaccinated patients, only 40% do not contract the flu.
If a patient is known to have contracted the flu, what is the probability that they were unvaccinated?
The Correct Answer and Explanation is:
To solve this problem, we use Bayes’ Theorem.
Given:
- P(V)=0.7P(V) = 0.7: Probability a patient is vaccinated.
- P(U)=0.3P(U) = 0.3: Probability a patient is unvaccinated (1 – 0.7).
- P(No Flu∣V)=0.9P(\text{No Flu} | V) = 0.9: Probability a vaccinated patient does not get the flu.
→ So, P(Flu∣V)=1−0.9=0.1P(\text{Flu} | V) = 1 – 0.9 = 0.1. - P(No Flu∣U)=0.4P(\text{No Flu} | U) = 0.4: Probability an unvaccinated patient does not get the flu.
→ So, P(Flu∣U)=1−0.4=0.6P(\text{Flu} | U) = 1 – 0.4 = 0.6.
We are asked to find:
What is the probability that a patient was unvaccinated given that they have contracted the flu?
This is P(U∣Flu)P(U | \text{Flu}), and we use Bayes’ Theorem: P(U∣Flu)=P(Flu∣U)⋅P(U)P(Flu)P(U | \text{Flu}) = \frac{P(\text{Flu} | U) \cdot P(U)}{P(\text{Flu})}
First, compute the total probability that a patient gets the flu: P(Flu)=P(Flu∣V)⋅P(V)+P(Flu∣U)⋅P(U)=(0.1)(0.7)+(0.6)(0.3)=0.07+0.18=0.25P(\text{Flu}) = P(\text{Flu} | V) \cdot P(V) + P(\text{Flu} | U) \cdot P(U) = (0.1)(0.7) + (0.6)(0.3) = 0.07 + 0.18 = 0.25
Now apply Bayes’ Theorem: P(U∣Flu)=(0.6)(0.3)0.25=0.180.25=0.72P(U | \text{Flu}) = \frac{(0.6)(0.3)}{0.25} = \frac{0.18}{0.25} = 0.72
Answer:
0.72\boxed{0.72}
Explanation
This problem involves conditional probability, where we are trying to determine the likelihood that a patient was unvaccinated given that they contracted the flu. Bayes’ Theorem is the standard approach in such scenarios.
Bayes’ Theorem allows us to “reverse” conditional probabilities. Instead of looking at the probability of getting the flu given vaccination status (which we’re given), we want to find the probability of being unvaccinated given a flu diagnosis.
The clinic data tells us that vaccinated patients are far less likely to get the flu—only 10% do. In contrast, 60% of unvaccinated patients catch the flu. While only 30% of patients are unvaccinated, they represent a disproportionate number of flu cases.
To find the probability that a flu patient is unvaccinated, we calculate:
- The overall flu rate (from both vaccinated and unvaccinated patients).
- How much of that flu rate is specifically from unvaccinated individuals.
- The proportion of flu cases that come from unvaccinated patients.
By Bayes’ Theorem, this gives us 72%. This means that if a patient shows up with the flu, there’s a 72% chance they weren’t vaccinated. This insight highlights the vaccine’s effectiveness—though a minority are unvaccinated, they make up most flu cases.
