I understand the first two parts of this three-part question, but am stuck on Part C:
The sensitivity V of a medical diagnostic test is the probability of detecting the disease (obtaining a positive test result) given that the patient actually has the disease. The specificity F, on the other hand, is the probability of a negative test result given that the patient is in fact healthy. Finally, by the prevalence P of a disease we meant the probability that a member of the population has the disease. Write HIV+ and T+ for the events that the patient is HIV positive and has a positive test result, respectively.
(A) Show that Pr(HIV+|T+) = (V x P) / [V x P + (1 - F) x (1 - P)].
(B) The sensitivity of the so-called ELISA (Enzyme Linked Immuno Sorbent Assay) test is 99 per cent, and its specificity 98.5 per cent. Based on the data in the table below, compute the probabilities of a patient being HIV positive if the test comes back positive in North America, Sub-Saharan Africa, and East Asia, respectively.
REGION Adult prevalence
Sub-Saharan Africa 0.059
East Asia 0.001
North America 0.008
(C) The events A and B are said to be conditionally independent given a third event C if Pr( A Intersection B | C ) = Pr(A|C) x Pr(B|C).
Compute the probabilities of a patient being HIV positive in North America, Sub-Saharan Africa, and East Asia respectively, given TWO positive test results, which are assumed to be conditionally independent.
I have completed parts A and B, and have calculated for B the following:
Pr(HIV+|T+) North America: 0.347368
Pr(HIV+|T+) Sub Saharan Africa: 0.805377
Pr(HIV+|T+) East Asia: 0.061972
Any help with Part C would be greatly appreciated. I was thinking of using the general form of Bayes Theorem, but am not sure how to set it up properly.