Engineering Statistics Question

[wolfmanzak]

Homework Statement

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?

Homework Equations

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

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.

 

[xur54]

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.

 

[xur54]

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.