Final answer:
To find the probability that a person has the virus given a positive test result, use Bayes' theorem with the relevant probabilities. Similarly, to find the probability that a person doesn't have the virus given a negative test result, apply Bayes' theorem with the probability of not testing positive.
Step-by-step explanation:
A certain virus infects 1 in every 2000 people, and we have a test that detects it with certain probabilities. To calculate the probability that a person has the virus given that they have tested positive, denoted as P(A | B), we'll use Bayes' theorem. Bayes' theorem relates the conditional and marginal probabilities of stochastic events.
First, we need to identify the following probabilities:
- P(A), the probability of being infected, which is 1/2000.
- P(B | A), the probability of testing positive if the person is infected, which is 96% or 0.96.
- P(B | not A), the probability of testing positive if the person is not infected (false positive rate), which is 4% or 0.04.
- P(not A), the probability of not being infected, which is 1 - (1/2000).
Bayes' theorem is given by:
P(A | B) = [P(B | A) * P(A)] / P(B), where P(B) is the total probability of testing positive. We calculate P(B) by:
P(B) = [P(B | A) * P(A)] + [P(B | not A) * P(not A)]
After calculating P(B), we plug it into the Bayes' theorem equation to find P(A | B).
To find the probability that a person does not have the virus given that they test negative, denoted as P(not A | not B), we again use Bayes' theorem in a similar fashion and calculate the probability of not testing positive:
P(not B) = 1 - P(B), which is the probability of any person testing negative. Then we use the ratio of the probability of a non-infected person testing negative to the total probability of testing negative to find P(not A | not B).