. Enzyme immunoassay (EIA) tests are used to screen blood specimens for the presence of antibodies to HIV, the virus that causes AIDS. Antibodies indicate the presence of the virus. The test is quite accurate but is not always correct. Here are approximate probabilities of positive and negative EIA outcomes when the blood tested does and does not actually contain antibodies to HIV:18

$1rn $1rn $1rn $1rn $1rn $1rn $1rn $1rn $1rn $1rn $1rn $1rn $1rn $1rn   Test Result

$1rn $1rn $1rn $1rn $1rn $1rn $1rn $1rn $1rn $1rn $1rn $1rn  +$1rn $1rn $1rn $1rn $1rn  −

Antibodies present$1rn $1rn $1rn $1rn $1rn $1rn    0.9985$1rn $1rn $1rn $1rn  0.0015

Antibodies absent$1rn $1rn $1rn $1rn $1rn $1rn $1rn 0.006$1rn $1rn $1rn $1rn    0.994

Suppose that 1% of a large population carries antibodies to HIV in their blood.

(a) Draw a tree diagram for selecting a person from this population (outcomes: antibodies present or absent) and for testing his

or her blood (outcomes: EIA positive or negative).

(b) What is the probability that the EIA is positive for a randomly chosen person from this population?

(c) What is the probability that a person has the antibody, given that the EIA test is positive? (Comment: This exercise illustrates a fact that is important when considering proposals for widespread testing for HIV, illegal drugs, or agents of biological warfare: if the condition being tested is uncommon in the population, many positives will be false positives.)


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