Question

A certain virus infects one in every 200 people. A test used to detect the virus in a person is positive 80% of the time when the person has the virus and 5% of the time when the person does not have the virus. (This 5% result is called a false positive.) Let A be the event "the person is infected" and B be the event "the person tests positive."

Using Bayes’ Theorem, when a person tests positive, determine the probability that the person is infected.

Answer 1

Using Bayes' Theorem, the probability that a person is infected when they test positive for the virus is approximately 9.1%.

Step-by-step explanation:

Using Bayes' Theorem, we can determine the probability that a person is infected when they test positive for the virus.

1. Let A be the event 'the person is infected' and B be the event 'the person tests positive'.
2. The probability of a person being infected, P(A), is 1/200 or 0.005 (since the virus infects one in every 200 people).
3. The probability of a positive test result given that the person is infected, P(B|A), is 0.8 or 80%.
4. The probability of a positive test result given that the person is not infected, P(B|A'), is 0.05 or 5% (this is the false positive rate).
5. We want to find the probability of a person being infected given that they test positive, P(A|B).
6. Applying Bayes' Theorem, we have P(A|B) = (P(B|A) * P(A)) / [(P(B|A) * P(A)) + (P(B|A') * P(A'))].
7. Substituting the given values, we get P(A|B) = (0.8 * 0.005) / [(0.8 * 0.005) + (0.05 * 0.995)].
8. Calculating this expression, we find that P(A|B) ≈ 0.091 or approximately 9.1%.