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 find the probability that a patient was unvaccinated given that they contracted the flu, we can use Bayes’ Theorem. Bayes’ Theorem helps us update the probability of a hypothesis (unvaccinated) based on new evidence (the patient contracted the flu). The formula for Bayes’ Theorem is:
[
P(\text{Unvaccinated | Flu}) = \frac{P(\text{Flu | Unvaccinated}) \cdot P(\text{Unvaccinated})}{P(\text{Flu})}
]
Where:
- ( P(\text{Unvaccinated | Flu}) ) is the probability the patient is unvaccinated given they contracted the flu (what we want to find).
- ( P(\text{Flu | Unvaccinated}) ) is the probability that a patient contracts the flu given they are unvaccinated.
- ( P(\text{Unvaccinated}) ) is the probability that a patient is unvaccinated.
- ( P(\text{Flu}) ) is the total probability that a patient contracts the flu, regardless of vaccination status.
Step 1: Identify the known probabilities
- ( P(\text{Vaccinated}) = 0.70 ) (70% of patients are vaccinated).
- ( P(\text{Unvaccinated}) = 1 – 0.70 = 0.30 ).
- Among vaccinated patients, 90% do not contract the flu, so 10% do. Hence, ( P(\text{Flu | Vaccinated}) = 0.10 ).
- Among unvaccinated patients, 40% do not contract the flu, so 60% do. Hence, ( P(\text{Flu | Unvaccinated}) = 0.60 ).
Step 2: Calculate ( P(\text{Flu}) )
The total probability of contracting the flu can be found by considering both vaccinated and unvaccinated groups:
[
P(\text{Flu}) = P(\text{Flu | Vaccinated}) \cdot P(\text{Vaccinated}) + P(\text{Flu | Unvaccinated}) \cdot P(\text{Unvaccinated})
]
Substituting the values:
[
P(\text{Flu}) = (0.10 \cdot 0.70) + (0.60 \cdot 0.30) = 0.07 + 0.18 = 0.25
]
Step 3: Apply Bayes’ Theorem
Now, we can plug the values into Bayes’ Theorem:
[
P(\text{Unvaccinated | Flu}) = \frac{0.60 \cdot 0.30}{0.25} = \frac{0.18}{0.25} = 0.72
]
Conclusion:
The probability that a patient who contracted the flu was unvaccinated is 0.72, or 72%.
Explanation:
This result shows that, given the flu, it is more likely that the patient was unvaccinated. This is because unvaccinated patients have a much higher probability of contracting the flu (60%) compared to vaccinated patients (10%). The prior probability of being unvaccinated (30%) combined with the higher likelihood of contracting the flu among unvaccinated individuals increases the conditional probability to 72%.