Your friend told you that he has been found as HIV-positive in a test recently.
ID: 3220761 • Letter: Y
Question
Your friend told you that he has been found as HIV-positive in a test recently. He is extreme worrying he will die from AIDS some years later because the test is very accurate that the sensitivity and specificity are 99.8% and 99.4%, respectively. However, the probability you know that he actually is an HIV-positive should be much lower than 99%. From the government website, you find that about 0.1% people in the area he live are HIV-positive.
(a) Calculate the probability that your friend is actually an HIV-positive given that he has been found as HIV-positive in the test.
(b) Explain why the calculated probability in (a) is much lower than the sensitivity and specificity.
Explanation / Answer
P represents positive result and I represents infected
P(P|I) = 0.998, P(P|I') = 0.002
P(P'|I') = 0.994
P(I) = 0.001
P(I') = 0.999
(A)
Required probability, P(I|P) = P(P|I)*P(I)/(P(P|I)*P(I) + P(P|I')*P(I'))
P(I|P) = 0.998*0.001/(0.998*0.001 + 0.002*0.999) = 0.3331
Hence 0.3311 is the probability that your friend is actually an HIV-positive given that he has been found as HIV-positive in the test.
(B)
The above calculated probability is much less than the sensitivity and specificity because there is very less population which exhibit the disease also there is significanct chance of getting the test positive in non infected population.
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