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:
We are given the following probabilities:
- P(V)=0.70P(V) = 0.70: Probability that a patient is vaccinated.
- P(U)=0.30P(U) = 0.30: Probability that a patient is unvaccinated (1 − 0.70).
- P(No Flu∣V)=0.90P(\text{No Flu} \mid V) = 0.90 ⟹ P(Flu∣V)=0.10P(\text{Flu} \mid V) = 0.10
- P(No Flu∣U)=0.40P(\text{No Flu} \mid U) = 0.40 ⟹ P(Flu∣U)=0.60P(\text{Flu} \mid U) = 0.60
We are asked to find:
P(U∣Flu)P(U \mid \text{Flu}): The probability that a patient was unvaccinated given they contracted the flu.
This is a conditional probability problem. We will use Bayes’ Theorem: P(U∣Flu)=P(Flu∣U)⋅P(U)P(Flu)P(U \mid \text{Flu}) = \frac{P(\text{Flu} \mid U) \cdot P(U)}{P(\text{Flu})}
We already have:
- P(Flu∣U)=0.60P(\text{Flu} \mid U) = 0.60
- P(U)=0.30P(U) = 0.30
We need to compute P(Flu)P(\text{Flu}), the total probability that a patient contracts the flu. We use the Law of Total Probability: P(Flu)=P(Flu∣V)⋅P(V)+P(Flu∣U)⋅P(U)P(\text{Flu}) = P(\text{Flu} \mid V) \cdot P(V) + P(\text{Flu} \mid U) \cdot P(U) P(Flu)=(0.10)(0.70)+(0.60)(0.30)=0.07+0.18=0.25P(\text{Flu}) = (0.10)(0.70) + (0.60)(0.30) = 0.07 + 0.18 = 0.25
Now plug into Bayes’ Theorem: P(U∣Flu)=(0.60)(0.30)0.25=0.180.25=0.72P(U \mid \text{Flu}) = \frac{(0.60)(0.30)}{0.25} = \frac{0.18}{0.25} = 0.72
✅ Final Answer: 0.72 (or 72%)
🔍 Explanation (300+ words):
This problem is a classic application of Bayes’ Theorem, which helps us reverse conditional probabilities — specifically, finding the probability that a person belongs to a particular group given an outcome. In this case, we want to know the chance that a flu-infected person was unvaccinated.
We start by organizing the given data into probabilities. We know that 70% of the clinic’s patients are vaccinated, and only 10% of these vaccinated individuals contract the flu. Among the 30% who are unvaccinated, a much higher 60% get the flu. We are trying to determine, if someone ends up getting the flu, how likely it is that they were unvaccinated.
Bayes’ Theorem uses the likelihood of the outcome for each group, along with the prior probabilities of being in each group, to calculate the posterior probability — the probability of the group given the outcome.
The total probability of getting the flu is found by summing the contributions from both groups:
- 70% are vaccinated, with 10% flu rate ⇒ 0.07
- 30% are unvaccinated, with 60% flu rate ⇒ 0.18
So 25% of all patients get the flu. Of these flu cases, 0.18 come from the unvaccinated group. Dividing 0.18 by the total 0.25 gives 0.72 — meaning 72% of all flu cases come from the unvaccinated group.
This result emphasizes how effective the flu vaccine is. Even though only 30% of people are unvaccinated, they make up the vast majority of flu cases — demonstrating a key concept in public health: vaccines greatly reduce disease burden, not just in individuals but in populations.