1. The HIV test we consider is an enzyme-linked immunosorbent assay, commonly known as an ELISA. ELISA’s true positive rate, also referred to as sensitively, is estimated as 𝑃(𝐸𝐿𝐼𝑆𝐴 𝑖𝑠 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒|𝑃𝑒𝑟𝑠𝑜𝑛 𝑡𝑒𝑠𝑡𝑒𝑑 ℎ𝑎𝑠 𝐻𝐼𝑉 ) = 0.93. Its true negative rate, also referred to as specificity, is estimated as 𝑃(𝐸𝐿𝐼𝑆𝐴 𝑖𝑠 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒|𝑃𝑒𝑟𝑠𝑜𝑛 𝑡𝑒𝑠𝑡𝑒𝑑 𝑛𝑜 𝐻𝐼𝑉 ) = 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. 𝑃(𝑃𝑒𝑟𝑠𝑜𝑛 𝑡𝑒𝑠𝑡𝑒𝑑 ℎ𝑎𝑠 𝐻𝐼𝑉) = 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)

To answer these questions, we will use Bayes' theorem, which is a fundamental theorem in the field of probability and statistics that describes the probability of an event based on prior knowledge of conditions that might be related to the event.

i. Compute the probability that someone has HIV if ELISA tests positive.

This is a classic application of Bayes' theorem. The formula is:

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 event A and event B respectively

In this case, we want to find P(HIV|ELISA+), the probability of having HIV given a positive ELISA test. We know that:

• P(ELISA+|HIV) = 0.93 (the probability of a positive ELISA test given HIV)
• P(HIV) = 0.00148 (the prevalence of HIV)
• P(ELISA+) = P(ELISA+|HIV) * P(HIV) + P(ELISA+|no HIV) * P(no HIV) = 0.93 * 0.00148 + (1 - 0.99) * (1 - 0.00148) (the total probability of a positive ELISA test)

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 simply 1 - P(HIV|ELISA+), the complement of the previous answer.

iii. What is the probability of being HIV positive if the second ELISA test comes back positive?

Assuming the tests are independent, the probability of having HIV given two positive ELISA tests is the same as the probability of having HIV given one positive ELISA test, which we calculated in part i.

iv. What is the probability that one has HIV after testing positive three times on the ELISA test?

Again, assuming the tests are independent, the probability of having HIV given three positive ELISA tests is the same as the probability of having HIV given one positive ELISA test, which we calculated in part i.