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.
The correct answer and explanation is :
In this scenario, we can calculate the probability that a patient does not contract the flu, given their vaccination status.
Given:
- 70% of patients are vaccinated against the flu. This means ( P(V) = 0.7 ), where ( V ) is the event that a patient is vaccinated.
- 30% of patients are unvaccinated, so ( P(U) = 0.3 ), where ( U ) is the event that a patient is unvaccinated.
- Among vaccinated patients, 90% do not contract the flu. This means the probability of not contracting the flu, given that the patient is vaccinated, is ( P(\text{No Flu} | V) = 0.9 ).
- Among unvaccinated patients, 40% do not contract the flu, so the probability of not contracting the flu, given that the patient is unvaccinated, is ( P(\text{No Flu} | U) = 0.4 ).
To find:
The total probability that a randomly selected patient does not contract the flu, which can be written as ( P(\text{No Flu}) ).
We can use the law of total probability:
[
P(\text{No Flu}) = P(\text{No Flu} | V)P(V) + P(\text{No Flu} | U)P(U)
]
Substituting the given values:
[
P(\text{No Flu}) = (0.9)(0.7) + (0.4)(0.3)
]
[
P(\text{No Flu}) = 0.63 + 0.12
]
[
P(\text{No Flu}) = 0.75
]
Conclusion:
The probability that a randomly selected patient does not contract the flu is 0.75, or 75%.
Explanation:
This problem uses conditional probabilities and the law of total probability. The total probability is calculated by considering both vaccinated and unvaccinated patients separately, then combining the results using their respective probabilities. By weighting the probabilities of not contracting the flu based on vaccination status (with 70% vaccinated and 30% unvaccinated), we can calculate the overall chance that a patient does not contract the flu. This total probability accounts for the fact that vaccinated patients have a much higher chance of avoiding the flu than unvaccinated patients.