Question

The New York State Health Department reports a 10% rate of the HIV virus for the "at-risk" population. Under certain conditions, a preliminary screening test for the HIV virus is correct 95% of the time. (Subjects are not told that they are HIV infected until additional tests verify the results.) If someone is randomly selected from the at-risk population, what is the probability that they have the HIV virus if it is known that they have tested positive in the initial screening?

Answer 1

The probability of having HIV after a positive preliminary screening test, given a 10% infection rate in the at-risk population and 95% test accuracy, is approximately 95%. This calculation accounts for a 0.5% false positive rate due to potential cross-reacting antibodies from other viruses.

Step-by-step explanation:

Calculating the Probability of Having HIV After a Positive Test

The probability of having HIV given a positive screening test can be calculated using Bayes' theorem. In the given scenario, there's a 10% HIV rate in the 'at-risk' population and the screening test is 95% accurate. However, because tests can give false positives, we must account for the 0.5% false positive rate that can occur due to cross-reacting antibodies from other viruses.

To solve this, we can use the formula:

P(HIV|Positive) = [P(Positive|HIV) * P(HIV)] / [P(Positive|HIV) * P(HIV) + P(False Positive) * (1 - P(HIV))]

Substituting the values we get:

P(HIV|Positive) = (0.95 * 0.1) / (0.95 * 0.1 + 0.005 * 0.9) = 0.095 / (0.095 + 0.0045) = 0.095 / 0.0995 ≈ 0.95 or 95%.

Thus, if someone from the 'at-risk' population tests positive, there is approximately a 95% chance they have the HIV virus, assuming the preliminary test result is positive.