Probability of H.I.V. - Math Central

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Question from Danielle, a student:

A medical test detects H.I.V. Among those who have H.I.V., the test will detect the disease with probability 0.95; among those who do NOT have H.I.V., the test will falsely claim that H.I.V. is present with probability 0.0125. Among those who take this test, 4% have H.I.V.

The test is given to Lucille, and indicates that she has H.I.V. What is the probability that Lucille actually has H.I.V.?

Hi Danielle.

How many ways can the test give her the result that H.I.V. is present?

(a) Lucille has H.I.V. and the test detects it.
(b) Lucille does not have H.I.V., but the test gives a false positive result.

4% of the people who take the test actually have H.I.V. Among those, the test detects it 95% of the time. So the chances that (a) happens is 4% x 95%.

There is a 96% chance that she doesn't have H.I.V., but when uninfected people are tested, 1.25% of them get false positive results. So the chances that (b) happens is 96% x 1.25%.

The question asks what is the probability that (a) happens rather than (b).

Can you finish the question now?
Stephen La Rocque.

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