Question

Blood donors are usually screened for HIV, both for the safety of the blood supply and for the benefit of the donor. The test, called an ELISA, tests positive 97.5% of the time if the donor actually has HIV. If the donor does not have HIV, the ELISA test correctly indicates that the person does not have the disease 92.6% of the time. About 0.2% of college students have HIV. After a blood drive at a college, the lab calls and tells a student that the test has indicated he has HIV. Use what you have learned in this lesson to determine the probability that the student actually has HIV. Because the prevalence of HIV is so small, it may be better to use 100,000 as your total population. A B Considering the answers to questions 1 and 2, should you be more concerned about a positive test for a rare disease or a common disease? Explain your answer.

Answer 1

Answer: To determine the probability that the student actually has HIV, we can use Bayes' theorem. Let A be the event that the student has HIV, and B be the event that the ELISA test indicates that the student has HIV. We want to find P(A|B), the probability that the student actually has HIV given that the test is positive.

From the problem, we are given:

P(B|A) = 0.975, the probability of a positive test given that the student has HIV.

P(B|A') = 0.074, the probability of a positive test given that the student does not have HIV.

P(A) = 0.002, the probability that a college student has HIV.

We can calculate P(A'), the probability that the student does not have HIV, as:

P(A') = 1 - P(A) = 1 - 0.002 = 0.998

Using Bayes' theorem, we have:

P(A|B) = P(B|A) * P(A) / [P(B|A) * P(A) + P(B|A') * P(A')]

Plugging in the values we have:

P(A|B) = 0.975 * 0.002 / [0.975 * 0.002 + 0.074 * 0.998]

= 0.025 / 0.0984

= 0.254

Therefore, the probability that the student actually has HIV given a positive test result is only about 25.4%.

Regarding the second question, we should be more concerned about a positive test for a rare disease. In this case, the prevalence of HIV is only 0.2%, which means that most people who test positive for HIV actually do not have the disease. On the other hand, for a common disease that affects a large proportion of the population, a positive test result is more likely to be accurate. Therefore, a positive test result for a rare disease is more concerning as it may lead to unnecessary treatment and anxiety for the patient.