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 who contracted the flu was unvaccinated, we use Bayes’ Theorem.
Step 1: Define Events
- Let VV be the event that a patient is vaccinated.
- Let UU be the event that a patient is unvaccinated.
- Let FF be the event that a patient contracts the flu.
From the problem:
- P(V)=0.7P(V) = 0.7, so P(U)=1−0.7=0.3P(U) = 1 – 0.7 = 0.3.
- The probability of contracting the flu given vaccination:
P(F∣V)=1−0.9=0.1P(F | V) = 1 – 0.9 = 0.1. - The probability of contracting the flu given no vaccination:
P(F∣U)=1−0.4=0.6P(F | U) = 1 – 0.4 = 0.6.
Step 2: Find the Total Probability of Contracting the Flu
Using the Law of Total Probability: P(F)=P(F∣V)P(V)+P(F∣U)P(U)P(F) = P(F | V) P(V) + P(F | U) P(U) P(F)=(0.1×0.7)+(0.6×0.3)P(F) = (0.1 \times 0.7) + (0.6 \times 0.3) P(F)=0.07+0.18=0.25P(F) = 0.07 + 0.18 = 0.25
Step 3: Apply Bayes’ Theorem
We need to find P(U∣F)P(U | F): P(U∣F)=P(F∣U)P(U)P(F)P(U | F) = \frac{P(F | U) P(U)}{P(F)} P(U∣F)=(0.6×0.3)0.25P(U | F) = \frac{(0.6 \times 0.3)}{0.25} P(U∣F)=0.180.25=0.72P(U | F) = \frac{0.18}{0.25} = 0.72
Conclusion
If a patient has the flu, the probability that they were unvaccinated is 0.72 (or 72%).
Explanation
This result shows that even though only 30% of the clinic’s patients are unvaccinated, they make up a disproportionately high percentage of flu cases due to their higher infection rate. This highlights the effectiveness of vaccines in preventing the flu.
Now, I will generate an image representing this probability scenario.
Here is the infographic illustrating the probability scenario. It visually represents the vaccination rates, flu contraction probabilities, and the final result that 72% of flu cases come from unvaccinated patients. Let me know if you need any modifications!
