# A probability question

• Aug 22nd 2012, 02:25 PM
bryce09
A probability question
I am trying to revise for a test, and am working through some prep questions. I am struggling on this one, could anyone help?

Suppose there is a medical diagnostic test for a disease. The sensitivity of the
test is .95. This means that if a person has the disease, the probability that the
test gives a positive response is .95. The specificity of the test is .90. This
means that if a person does not have the disease, the probability that the test
gives a negative response is .90, or that the false positive rate of the test is .10.
In the population, 1% of the people have the disease.
What is the probability that a person tested has the disease, given the results of the test is positive? Let
D be the event "the person has the disease" and let T be the event "the test
gives a positive result."

Any help would be much appreciated.
• Aug 22nd 2012, 02:54 PM
MaxJasper
Re: A probability question
n = normal
s = sick
+ = test positive
- = test negative

Pr(n+) = .1 * .99 = .099
Pr(n-) = .90 * .99 = .891
Pr(s+) = .95 * .01 = .0095
Pr(s-) = .05 * .01 = .0005

Pr(s|+) = .0095/(.099+.0095) = .08756 (=8.756%)
• Aug 22nd 2012, 03:11 PM
Plato
Re: A probability question
Quote:

Originally Posted by bryce09
Suppose there is a medical diagnostic test for a disease. The sensitivity of the
test is .95. This means that if a person has the disease, the probability that the
test gives a positive response is .95. The specificity of the test is .90. This
means that if a person does not have the disease, the probability that the test
gives a negative response is .90, or that the false positive rate of the test is .10.
In the population, 1% of the people have the disease.
What is the probability that a person tested has the disease, given the results of the test is positive? Let
D be the event "the person has the disease" and let T be the event "the test
gives a positive result."

I think that I answered this elsewhere. Let $\displaystyle D$ means a person has the disease and $\displaystyle D^c$ means a person does not have the disease. You are given: $\displaystyle P(+|D)=0.95,~P(-|D^c)=0.90,~\&~P(D)=0.01$.
From that we conclude $\displaystyle P(-|D)=0.05,~P(+|D^c)=0.10,~\&~P(D^c)=0.99~.$
Now $\displaystyle P(+)=P(+|D)P(D)+P(+|D^c)P(D^c)$

The question is $\displaystyle P(D|+)=\frac{P(D\cap +)}{P(+)}~.$
• Aug 22nd 2012, 05:54 PM
bryce09
Re: A probability question