the probability of a person having the virus given that they tested positive is approximately 2.21%.
Given Information:
- Virus Infection Rate: 1 in 400 people (infected = 1/400, not infected = 399/400)
- Test Positivity:
- Positive if infected: 90% (true positive)
- Positive if not infected: 10% (false positive)
Events:
- A: Person is infected (probability = 1/400)
- B: Person tests positive (probabilities depend on being infected or not)
Part (a): Probability of Infection Given Positive Test
We want to find P(A|B), the probability of someone being infected (event A) given that they tested positive (event B). This is a Bayes' theorem application.
Using Bayes' theorem:
P(A|B) = P(B|A) * P(A) / P(B)
where:
- P(A|B) is the probability of being infected given a positive test (what we want to find)
- P(B|A) is the probability of testing positive given someone is infected (true positive rate = 90%)
- P(A) is the prior probability of being infected (1/400)
- P(B) is the overall probability of someone testing positive (need to calculate this)
Calculating P(B):
P(B) is the probability of someone testing positive, which can happen due to a true positive (infected and test positive) or a false positive (not infected but test positive).
P(B) = P(B|A) * P(A) + P(B|not A) * P(not A)
where:
- P(B|A) is the true positive rate (already mentioned as 90%)
- P(A) is the prior probability of being infected (1/400)
- P(B|not A) is the false positive rate (10%)
- P(not A) is the probability of not being infected (399/400)
Plugging in the values:
P(B) = 0.9 * (1/400) + 0.1 * (399/400) = 0.102
Back to Bayes' theorem:
Now we have all the values to solve for P(A|B):
P(A|B) = (0.9 * 1/400) / 0.102 = 0.0221
Therefore, the probability of a person having the virus given that they tested positive is approximately 2.21%.