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Engineering Statistics Question

  1. Nov 3, 2012 #1
    1. The problem statement, all variables and given/known data
    The proportion of people in a given community who have a certain disease is 0.005. A test is available to diagnose the disease. If a person has the disease, the probability that the test will produce a positive signal is 0.98. If a person does not have the disease, the probability that the test will produce a positive signal is 0.02.

    If a person tests negative, what is the probability that the person actually has the disease?

    2. Relevant equations
    I'm at a loss for the relevant equation. I've scanned through my book several times.


    3. The attempt at a solution
    I set the problem up as follows:
    Given the following:
    Probability that one has the disease: P(D)=.005
    Probability that given one has the disease, they test positive: P(+|D)=0.98
    Probability that given one does not have the disease, they test positive: P(+|D^c)=.02

    I've used the notation P(-) for test is negative which is equal to P(+^c)

    I'm trying to find P(D|-) based on the problem statement. I can't find any way to relate this to what I've been given. The solution says to use P(+|D^c)P(D)=.02*.005=1.0E-4 but I have no idea where that relation came from or where to find a proof of such online.

    If anyone could explain to me why P(D|-)=P(+|D^c)P(D) it would really help my understanding of the problem.
     
  2. jcsd
  3. Nov 3, 2012 #2
    Well I agree with the answer at least. Intuitive word explanation: Trying to find how many people who actually have the disease are missed. We know .005 are diseased. If you could test that group, .98*.005=.0049 would be confirmed. (1-.98)*.005=.0001=1e-4 are missed. They have it but got a false test.
    I'm still working on the proof. The solutions are wrong sometimes, but in this case I would go with the solutions over me. Hopefully I'll get the same work they did.
     
    Last edited: Nov 3, 2012
  4. Nov 3, 2012 #3
    First off, for the notation you're using, the | operator is not commutative if I understand it correctly. So P(-|D)=0.02 does not equal P(D|-).
    All I can figure out is that maybe there is a law that for good tests P(-|D)=P(+|D^c). To me that doesn't seem reasonable. [edit: actually now it seems somewhat reasonable but still can't help too much]
    Sorry I couldn't help you more. Maybe someone else can.
     
    Last edited: Nov 3, 2012
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