Test for HIV. There's a false positive rate of 0.025 and a false negative rate of 0.08. Let's say a particular patient has a probability of testing positive for HIV of 0.005. The patient gets tested and it's positive. What are the chances that the patient actually has HIV?

Kenya Leonard 2022-07-23 Answered
Test for HIV. There's a false positive rate of 0.025 and a false negative rate of 0.08. Let's say a particular patient has a probability of testing positive for HIV of 0.005. The patient gets tested and it's positive. What are the chances that the patient actually has HIV?
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Answers (1)

Brendon Bentley
Answered 2022-07-24 Author has 11 answers
This is a typical application of Bayes formula,
P ( B | A ) = P ( A | B ) P ( B ) / ( P ( A | B ) P ( B ) + P ( A | B ¯ ) P ( B ¯ ) ) .
Here A is testing positive, B is having HIV, and B ¯ is not having HIV. P ( A | B ) is the true positive rate, the complement of the fpr .08. P ( A | B ¯ ) is the false positive rate .025. Substituting the numbers you give, P ( B | A ) = .92 .005 / ( .92 .005 + .025 .995 ) = 0.1560645.
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