The HIV test we consider is an enzyme-linked immunosorbent assay, commonly known as an ELISA. ELISAโs true positive rate, also referred to as sensitively, is estimated as ๐(๐ธ๐ฟ๐ผ๐๐ด ๐๐ ๐๐๐ ๐๐ก๐๐ฃ๐|๐๐๐๐ ๐๐ ๐ก๐๐ ๐ก๐๐ โ๐๐ ๐ป๐ผ๐ ) = 0.93. Its true negative rate, also referred to as specificity, is estimated as ๐(๐ธ๐ฟ๐ผ๐๐ด ๐๐ ๐๐๐๐๐ก๐๐ฃ๐|๐๐๐๐ ๐๐ ๐ก๐๐ ๐ก๐๐ ๐๐ ๐ป๐ผ๐ ) = 0.99. Also, the prevalence of HIV in the overall population, which is estimated to be 1.48 out of every 1000 American adults. i.e. ๐(๐๐๐๐ ๐๐ ๐ก๐๐ ๐ก๐๐ โ๐๐ ๐ป๐ผ๐) = 0.00148. i. Compute the probability that someone has HIV if ELISA tests positive. ii. What is the probability that someone who tests positive does not have HIV? iii. What is the probability of being HIV positive if the second ELISA test comes back positive? iv. What is the probability that one has HIV after testing positive three times on the ELISA test? (Assume that all three ELISAs are independent)
Question
The HIV test we consider is an enzyme-linked immunosorbent assay, commonly known as an ELISA. ELISAโs true positive rate, also referred to as sensitively, is estimated as ๐(๐ธ๐ฟ๐ผ๐๐ด ๐๐ ๐๐๐ ๐๐ก๐๐ฃ๐|๐๐๐๐ ๐๐ ๐ก๐๐ ๐ก๐๐ โ๐๐ ๐ป๐ผ๐ ) = 0.93. Its true negative rate, also referred to as specificity, is estimated as ๐(๐ธ๐ฟ๐ผ๐๐ด ๐๐ ๐๐๐๐๐ก๐๐ฃ๐|๐๐๐๐ ๐๐ ๐ก๐๐ ๐ก๐๐ ๐๐ ๐ป๐ผ๐ ) = 0.99. Also, the prevalence of HIV in the overall population, which is estimated to be 1.48 out of every 1000 American adults. i.e. ๐(๐๐๐๐ ๐๐ ๐ก๐๐ ๐ก๐๐ โ๐๐ ๐ป๐ผ๐) = 0.00148. i. Compute the probability that someone has HIV if ELISA tests positive. ii. What is the probability that someone who tests positive does not have HIV? iii. What is the probability of being HIV positive if the second ELISA test comes back positive? iv. What is the probability that one has HIV after testing positive three times on the ELISA test? (Assume that all three ELISAs are independent)
Solution
To solve these problems, we will use Bayes' theorem, which is a fundamental theorem in the field of probability and statistics. Bayes' theorem is used to update the probability of a hypothesis based on evidence. It is given by the formula:
P(A|B) = P(B|A) * P(A) / P(B)
where:
- P(A|B) is the probability of event A given event B is true.
- P(B|A) is the probability of event B given event A is true.
- P(A) and P(B) are the probabilities of events A and B respectively.
i. Compute the probability that someone has HIV if ELISA tests positive.
Let's denote:
- A as the event "Person tested has HIV"
- B as the event "ELISA is positive"
We are asked to find P(A|B), the probability that someone has HIV given that the ELISA test is positive.
From the problem, we know that:
- P(B|A) = 0.93 (the probability that ELISA is positive given the person has HIV)
- P(A) = 0.00148 (the prevalence of HIV in the population)
- P(B) can be calculated as P(B|A)*P(A) + P(B|~A)*P(~A), where ~A is the event "Person tested does not have HIV". We know that P(B|~A) = 1 - 0.99 = 0.01 (the probability that ELISA is positive given the person does not have HIV), and P(~A) = 1 - P(A).
Substituting these values into Bayes' theorem gives us the answer.
ii. What is the probability that someone who tests positive does not have HIV?
This is asking for P(~A|B), the probability that someone does not have HIV given that the ELISA test is positive. This can be calculated as 1 - P(A|B), which we found in the previous step.
iii. What is the probability of being HIV positive if the second ELISA test comes back positive?
If the tests are independent, the probability of being HIV positive given two positive tests is the same as the probability of being HIV positive given one positive test, which is P(A|B).
iv. What is the probability that one has HIV after testing positive three times on the ELISA test?
Similarly, if the tests are independent, the probability of being HIV positive given three positive tests is the same as the probability of being HIV positive given one positive test, which is P(A|B).
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