Final answer:
To find the probability that an individual who tested positive actually had the virus, we use Bayes' Theorem and given data about the test's false positive rate, false negative rate, and prevalence of the virus. After calculation, the probability is approximately 16.8%, which is option (b) 0.168.
Step-by-step explanation:
The question is asking to compute the probability that an individual who tested positive actually had the virus given an antibody test with a 5% false positive rate and a 0% false negative rate, with the prevalence of the virus being 1% in the population. This can be solved using Bayes' Theorem.
Let's define the following events:
- A - The event that an individual has the virus.
- B - The event that an individual tests positive for the virus.
Given the data:
- P(A) = 0.01 (1% of the population has the virus)
- P(B|A) = 1 (If you have the virus, you will test positive, since the false negative rate is 0%)
- P(B|A') = 0.05 (If you don't have the virus, there is a 5% chance that you will still test positive, which is the false positive rate)
- We want to find P(A|B), the probability that a person has the virus given that they tested positive.
Bayes' Theorem states that:
P(A|B) = (P(B|A) * P(A)) / (P(B|A) * P(A) + P(B|A') * P(A'))
Where P(A') = 1 - P(A) is the probability of not having the virus.
Plugging in the values we get:
P(A|B) = (1 * 0.01) / ((1 * 0.01) + (0.05 * (1 - 0.01)))
After calculating the above expression, we find that:
P(A|B) ≈ 0.168, or 16.8%
Therefore, the probability that an individual who tested positive actually had the virus is approximately 0.168 or 16.8%. Thus, the correct answer from the provided options is (b) 0.168.