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Math Help - Given a positive test result, find the probability that test taker has the disease.

  1. #1
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    Given a positive test result, find the probability that test taker has the disease.

    Hi All,

    I have a probability problem that I've chased around until I'm sick of looking at it. Here goes.

    The incidence of a disease is .012. A test, with sensitivity = .86, and specificity =.88. Giben a positive test result, what is the probability that the test taker has the disease.

    The solution, that I can't get to, is 8%.

    While I'm not sure I've dine anything right, I am sure that I'm having some trouble applying the specificity and sensitivity.

    Thanks for lookin in!
    Last edited by mr fantastic; May 20th 2010 at 05:42 AM. Reason: Re-titled.
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  2. #2
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    You have to use Bayes' Theorem:

    pr(D)= the probability that the person has the disease = 0.012
    pr(H)= the probability that the person does not have the disease = 0.988
    pr(+|D)= the probability that the person will test positive for the disease given that they have it(sensitivity) = 0.86
    pr(+|H)= the probability that the test will return positive given that the person does not have the disease (1-specificity)=0.12

    Using Bayes' Theorem:

    pr(D|+)=\frac{pr(+|D)*pr(D)}{(pr(+|D)*pr(D))+(pr(+  |H)*pr(H))}

    pr(D|+)=\frac{0.86*0.012}{(0.86*0.012)+(0.12*0.998  )}= \frac{0.01032}{0.01032+0.11976}= \frac{0.01032}{0.13008}=\frac{1032}{13008}= \frac{43}{542}\approx 0.079 \approx 8\%
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  3. #3
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    Thanks!
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