# Disease Probability

• September 25th 2007, 09:15 AM
DivideBy0
Disease Probability
6
One percent of the population suffers from a certain disease. There is a blood test for this disease, and it is 99% accurate, in other words, the probability that it gives the correct answer is 0.99, regardless of whether the person is sick or healthy. A person takes the blood test, and the result says that she has the disease. What is the probability that she actually has the disease?

(A) 0.99%
(B) 25%
(C) 50%
(D) 75%
(E) 98%

I thought it would just be 0.1*0.99 = 0.0099 but it appears that this is not the case.
• September 25th 2007, 12:16 PM
Plato
From the given you can see that:
$P(D) = 0.01$ probability of having the disease.
$P(D^c) = 0.99$ probability of not having the disease.
$P( + |D) = 0.99$ probability of a positive test given the disease.
$P( + |D^c) = 0.01$ probability of a positive test given no disease.

The probability of a positive test is:
$P( + ) = P( + D) + P( + D^c ) = P( + |D)P(D) + P( + |D^c )P(D^c )$

What the question asks you to find is:
$P(D| + ) = \frac{{P(D + )}}{{P( + )}} = \frac{{P( + |D)P(D)}}{{P( + )}}$.

Can you finish?