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Expert: - 5/16/2008


Question
A test correctly identifies a disease in 95% of people who have it. It correctly identifies no disease in 94% of people who do not have it. In the population, 3% of the people have the disease. What is the probability that you have the disease if you tested positive?


Answer

Venn Diagram
Questioner:   Wendy
Category:  Advanced Math
Private:  No
 
Subject:  Probability and Statistics
Question:  A test correctly identifies a disease in 95% of people who have it. It correctly identifies no disease in 94% of people who do not have it. In the population, 3% of the people have the disease. What is the probability that you have the disease if you tested positive?
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Hi, Wendy,

Let D and T represent these sets:

D  = A person has the disease.
D' = A person is well.

T = the test is positive for the disease.
T'= the test is negative for the disease.

You are told:

p(D) = 0.03

p(T, if D) = 0.95

p(T', if D') = 0.94

There are four sections: [See attached two-ring diagram.]

p(D and T) : p(#3)

p(D and T') : p(#2)

p(D' and T) : p(#4)

p(D' and T') : p(#1)

which add up to 1.000
...........................................
A test correctly identifies a disease in 95% of people who have it.  

That means  #3 is 95% of #2+#3

But #2 + #3 = 0.03 = % of people with disease.
And #1 + #4 = 0.97 = % of people without disease.

So  #3 = 0.03 * 0.95 = 0.0285 = % of people with disease and positive test.
And #2 = 0.0015 = % of people with disease and NEGAtive test.
..................................
It correctly identifies no disease in 94% of people who do not have it.

That means #1 is 94% of #1 + #4
And #1 + #4 = 0.97 = % of people without disease.
So #1 = 0.97 * 0.94 = 0.9118 = % of people withOUT disease and negative test.
And #4 = 0.97 * 0.06 =  0.0582 =   % of people with disease and positive test.

Summary:

#1 = 0.9118
#2 = 0.0015
#3 = 0.0285
#4 = 0.0582
-------------
    1.0000

Now we are ready for your question:

What is the probability that you have the disease if you tested positive?

That means: what is p(D and T, given T)

That is  p(D and T)/p(T) = p(#3)/[p(#3) + p(#4)]

= 0.0285/[0.0285 + 0.0582]

= 0.0285/[0.0867] = 0.3287

    Questioner's Rating
    Rating(1-10)Knowledgeability = 10Clarity of Response = 10Politeness = 10
    CommentThank You. I hadn't heard or thought about a Venn Diagram since I was in grade school. We didn't realize we were doing advance math then.


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Paul Klarreich

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I can answer questions in basic to advanced algebra (theory of equations, complex numbers), precalculus (functions, graphs, exponential, logarithmic, and trigonometric functions and identities), basic probability, and finite mathematics, including mathematical induction. I can also try (but not guarantee) to answer questions on Abstract Algebra -- groups, rings, etc. and Analysis -- sequences, limits, continuity. I won't understand specialized engineering or business jargon.

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I taught at a two-year college for 25 years, including all subjects from algebra to third-semester calculus.

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