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Math Help - Help with false positive question

  1. #1
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    Help with false positive question

    A blood test is 99% effective in detecting a certain disease when the
    disease is present. However, the test also yields a false positive result
    for 2% of healthy patients tested. Suppose that 0.5% of the population
    have the disease.

    (a) What proportion of healthy patients are correctly diagnosed?

    (b) What is the unconditional (or marginal) probability of a positive
    test result?

    (c) What is the probability of having the disease given a positive test
    result?
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  2. #2
    MHF Contributor
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    Just in case a picture helps...

    ... with W = well, I = ill, P = testing positive, N = testing negative...



    See here Conditional probability - Wikipedia, the free encyclopedia for a more formal treatment of a very similar problem (see under the heading 'An example'), but from the tree you can see that

    (b) is (398 + 99) / 20,000

    and (c) is 99 / (398 + 99)

    (a) is disconcertingly more obvious and (if I'm not mis-reading it) just 98%. Hope this helps.


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  4. #4
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    Quote Originally Posted by Turloughmack View Post
    A blood test is 99% effective in detecting a certain disease when the
    disease is present. However, the test also yields a false positive result
    for 2% of healthy patients tested. Suppose that 0.5% of the population
    have the disease.

    (a) What proportion of healthy patients are correctly diagnosed?

    (b) What is the unconditional (or marginal) probability of a positive
    test result?

    (c) What is the probability of having the disease given a positive test
    result?
    I often find it helpful to suppose there are say 1,000,000 patients, 999,500 are healthy and 500 have the disease

    19990 of the healthy test positive, and 495 of the diseased test positive.

    So for (a) the answer is (999500-19990)/999500=0.98

    For (b) (19990+495)/1000000

    For (c) ...

    CB
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