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 can use Bayes’ Theorem to solve this problem. Let’s define the events:
- ( V ) = Patient is vaccinated.
- ( U ) = Patient is unvaccinated.
- ( F ) = Patient contracts the flu.
From the problem statement:
- ( P(V) = 0.7 ), meaning 70% of patients are vaccinated.
- ( P(U) = 1 – P(V) = 0.3 ), meaning 30% are unvaccinated.
- ( P(F | V) = 1 – 0.9 = 0.1 ), meaning 10% of vaccinated patients contract the flu.
- ( P(F | U) = 1 – 0.4 = 0.6 ), meaning 60% of unvaccinated patients contract the flu.
We need to find ( P(U | F) ), the probability that a patient who contracted the flu was unvaccinated. Using Bayes’ Theorem:
[
P(U | F) = \frac{P(F | U) P(U)}{P(F)}
]
First, calculate ( P(F) ), the total probability that a patient contracts the flu, using the Law of Total Probability:
[
P(F) = P(F | V) P(V) + P(F | U) P(U)
]
[
P(F) = (0.1 \times 0.7) + (0.6 \times 0.3)
]
[
P(F) = 0.07 + 0.18 = 0.25
]
Now, compute ( P(U | F) ):
[
P(U | F) = \frac{(0.6 \times 0.3)}{0.25}
]
[
P(U | F) = \frac{0.18}{0.25} = 0.72
]
Final Answer:
The probability that a patient who contracted the flu was unvaccinated is 0.72 (or 72%).
Explanation:
The calculation is based on Bayes’ Theorem, which allows us to update probabilities based on given evidence. While only 30% of patients are unvaccinated, they are much more likely to contract the flu (60% vs. 10%). Since more flu cases come from unvaccinated patients, the likelihood that a flu patient was unvaccinated is 72%, even though they are a minority in the total population.