The probability that a patient has HIV is 0.001 and the diagnostic test for HIV can detect the virus with a probability of 0.98. Given that the chance of a false positive is 6%, what is the probability that a patient who has already tested positive really has HIV?

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ipa1rafd · Accepted Answer

A positive result can come in two ways: a patient with HIV and a correct result:   0.001  ⋅  0.98  =  0.00098 of the population, or a patient without HIV and a false positive. The fraction of false positives is   0.999  ⋅  0.06  =  0.05994. The false positives are then       0.05994    0.0094    ≈  61.2 times more than the infected patients.

gasavasiv · Accepted Answer

This is a classic example for the illustration of Bayes&#039; theorem. Let&#039;s first formulate the problem in formal terms. Let   D be the event that the person has the disease, then       D    c   denotes the event that the person doesn&#039;t have the disease. Let   Y be the event that the test gives the positive result (person has the disease as per the test diagnostic) and   N be the event that the test gives the negative result.Now let&#039;s write down the given information.  P  (  D  )  =  0.001  P  (  Y      |    D  )  =  0.98  P  (  Y      |        D    c    )  =  0.06We have to find   P  (  D      |    Y  ).Now we&#039;ll use Bayes&#039; theorem to find the required probability.   P  (  D      |    Y  )  =            P      (      Y              |            D      )      P      (      D      )              P      (      Y      )        P  (  Y  )  =  P  (  Y  ∩  (  D  ∪      D    c    )  )  =  P  (  Y  ∩  D  )  +  P  (  Y  ∩      D    c    )  =  P  (  Y      |    D  )  P  (  D  )  +  P  (  Y      |        D    c    )  P  (      D    c    )as   D and       D    c   are mutually exclusive events and together form a partition of the sample space. Using the given values, we have  P  (  Y  )  =  0.98  ×  0.001  +  0.06  ×  0.999  =  0.06092Therefore,  P  (  Y      |    D  )  =            0.98      ×      0.001        0.06092    =  0.016

The probability that a patient has HIV is 0.001 and the diagnostic test for HIV can detect the virus with a probability of 0.98. Given that the chance of a false positive is 6%, what is the probability that a patient who has already tested positive really has HIV?

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