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?
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a) The four values in the table represent the probabilities of different outcomes in the context of HIV testing:
1. 99.85% (0.9985) - This is the probability that a person with HIV antibodies will test positive for HIV. In other words, if a person actually has HIV antibodies, there is a 99.85% chance that the test will correctly identify them as HIV positive.
2. 0.6% (0.006) - This is the probability that a person without HIV antibodies will test positive for HIV. It represents the chance of a false-positive result, where the test incorrectly identifies someone without HIV antibodies as HIV positive.
3. 0.15% (0.0015) - This is the probability that a person with HIV antibodies will test negative for HIV. It represents the chance of a false-negative result, where the test fails to detect HIV antibodies in someone who actually has them.
4. 99.4% (0.994) - This is the probability that a person without HIV antibodies will test negative for HIV. In other words, if a person does not have HIV antibodies, there is a 99.4% chance that the test will correctly identify them as HIV negative.
b) To find the probability of a false-positive result, we need to consider the probability that a person without HIV antibodies (0.02 or 2% of the population) will test positive for HIV. From the table, we see that the probability is 0.6% (0.006), which means there is a 0.6% chance that a person without HIV will have a positive test result.
c) To find the probability of a false-negative result, we need to consider the probability that a person with HIV antibodies (also 2% of the population) will test negative for HIV. From the table, we see that the probability is 0.15% (0.0015), which means there is a 0.15% chance that a person with HIV will have a negative test result.
d) The probability that a person with HIV will have a positive test result is equivalent to the probability of a true-positive result, which is 99.85% (0.9985). This means that there is a 99.85% chance that a person with HIV antibodies will be correctly identified as HIV positive by the test.