The tests done to determine if someone is HIV positive are called Enzyme immunoassay or EIA tests. The test screens a blood sample for the presence of antibodies to HIV. Like most tests, this test is not perfect. The table below shows the approximate probabilities of positive and negative EIA tests when the blood does and does not actually contain the HIV antibodies. Long range studies have shown that only 2% of the population actually has the HIV antibodies.
Test Result Positive Negative
Antibodies Present 0.9985 0.0015
Antibodies Not Present 0.006 0.994
a) Explain in context all four values in the table. That is, what does each of them actually mean? I have 99.85%,0.6%,0.15% and 99.4%
Construct a tree diagram to answer the following questions. The first two choices of the tree should be whether or not the antibodies are present. The second choice should be the test result. (What is the probability split for the first choice?)
b) What is the probability that a person without HIV will have a test come out positive (this is called a false-positive)?
c) What is the probability that a person with HIV will have a test come out negative (this is called a false negative)?
d) What is the probability that a person with HIV will have a test come out positive?
a) The table contains four values:
- 0.9985 (99.85%): This represents the probability of a positive test result when the person actually has the HIV antibodies. In other words, when antibodies are present, there is a 99.85% chance that the test will accurately detect them.
- 0.006 (0.6%): This indicates the probability of a positive test result when the person does not have the HIV antibodies. It represents the false-positive rate, meaning the test incorrectly identifies the presence of HIV antibodies in 0.6% of individuals who do not actually have them.
- 0.0015 (0.15%): This represents the probability of a negative test result when the person does have the HIV antibodies. It indicates the false-negative rate, as the test fails to detect the antibodies in 0.15% of cases where they are present.
- 0.994 (99.4%): This is the probability of a negative test result when the person does not have the HIV antibodies. It shows that the test correctly identifies the absence of antibodies in 99.4% of cases.
b) To determine the probability of a false-positive, we look at the second row of the table, which represents cases where the antibodies are not present. So, the probability is 0.006, or 0.6%.
c) The probability of a false-negative can be found in the third column of the table, which corresponds to cases where the antibodies are actually present. Therefore, the probability is 0.0015, or 0.15%.
d) The probability of a true positive, or the likelihood of a person with HIV having a positive test result, is given in the first row of the table. So, the probability is 0.9985, or 99.85%.