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, which helps us find the probability of a condition (unvaccinated) given an outcome (contracted the flu).
Given Data:
- P(V)=0.70P(V) = 0.70: Probability a patient is vaccinated
- P(U)=0.30P(U) = 0.30: Probability a patient is unvaccinated
- P(No Flu∣V)=0.90P(\text{No Flu} | V) = 0.90 ⇒ P(Flu∣V)=0.10P(\text{Flu} | V) = 0.10
- P(No Flu∣U)=0.40P(\text{No Flu} | U) = 0.40 ⇒ P(Flu∣U)=0.60P(\text{Flu} | U) = 0.60
We are asked:
What is P(U∣Flu)P(U | \text{Flu})?
This is the probability a patient is unvaccinated given that they contracted the flu.
Step 1: Apply 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})}
We already know:
- P(Flu∣U)=0.60P(\text{Flu} | U) = 0.60
- P(U)=0.30P(U) = 0.30
We need to compute the total probability of contracting the flu, P(Flu)P(\text{Flu}), using the Law of Total Probability: P(Flu)=P(Flu∣V)⋅P(V)+P(Flu∣U)⋅P(U)P(\text{Flu}) = P(\text{Flu} | V) \cdot P(V) + P(\text{Flu} | U) \cdot P(U) P(Flu)=(0.10⋅0.70)+(0.60⋅0.30)=0.07+0.18=0.25P(\text{Flu}) = (0.10 \cdot 0.70) + (0.60 \cdot 0.30) = 0.07 + 0.18 = 0.25
Step 2: Plug into Bayes’ Theorem
P(U∣Flu)=0.60⋅0.300.25=0.180.25=0.72P(U | \text{Flu}) = \frac{0.60 \cdot 0.30}{0.25} = \frac{0.18}{0.25} = 0.72
✅ Final Answer:
0.72\boxed{0.72}
Explanation:
This means that if a patient is known to have contracted the flu, there is a 72% chance that they were unvaccinated. Even though only 30% of the clinic population is unvaccinated, they represent a disproportionately large part of flu cases because their flu risk (60%) is much higher than that of vaccinated individuals (10%). This reflects how effective vaccination is in preventing flu and how Bayes’ Theorem allows us to reverse conditions (from outcome to cause).