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) In the given table, the four values correspond to different probabilities:

1. 99.85% (0.9985): This is the probability that the EIA test will correctly detect the presence of HIV antibodies in a blood sample. In other words, if a person has HIV antibodies, there is a 99.85% chance that the test will correctly identify it as positive.

2. 0.6% (0.006): This is the probability of a false-positive result. It represents the likelihood that the test will incorrectly identify a person without HIV antibodies as positive. In other words, if a person does not have HIV antibodies, there is a 0.6% chance that the test will incorrectly show a positive result.

3. 0.15% (0.0015): This is the probability of a false-negative result. It represents the likelihood that the test will incorrectly identify a person with HIV antibodies as negative. In other words, if a person has HIV antibodies, there is a 0.15% chance that the test will incorrectly show a negative result.

4. 99.4% (0.994): This is the probability that the EIA test will correctly exclude the presence of HIV antibodies in a blood sample. In other words, if a person does not have HIV antibodies, there is a 99.4% chance that the test will correctly identify it as negative.

b) To find the probability of a false-positive result, we need to consider the scenario where the person does not have HIV antibodies but still gets a positive test result. From the table, we can see that the probability of a false-positive is 0.6% or 0.006.

c) To find the probability of a false-negative result, we need to consider the scenario where the person has HIV antibodies but still gets a negative test result. From the table, we can see that the probability of a false-negative is 0.15% or 0.0015.

d) To find the probability of a true-positive result (i.e., a person with HIV getting a positive test), we can subtract the false-negative probability from 100%. Therefore, the probability of a person with HIV having a positive test is 100% - 0.15% = 99.85% or 0.9985.