QUESTION

A particular heart disease has a prevalence of 1/1000 people. A test to detect this disease has a false positive rate of 5%. This means that the probability of getting a positive results GIVEN that you do NOT have the disease (that is, p(B|notA) is .05. Assume that the test diagnoses correctly every person who has the disease.

Q: What is the chance that a randomly selected person found to have a positive result actually has the disease?

(If you do NOT want to see the solution yet, do NOT scroll down. When you DO look at the solution below, please note that the html file is NOT displaying the Bayes theorem formula very well. Look in your notes if you are confused.)

 

 

 

 

 

 

SOLUTION

The formal Bayesian approach:

Let A = person has the disease

Let B = positive test

We wish to calculate p(A| B), It is vitally important that you understand that this is the probability sought.

Using Bayes’s theorem, we can compute this probability.

p(A| B) = p(B| A)p(A)

p(B| A)p(A) + p(B| not A)p(not A)

 

Of these probabilities, which do we know?

p(A) = .001 (This is given in the problem.)

p(notA) = .999 (Why? You compute this from knowing that p(A) = .001.)

p(B| notA) = .05 This is given to you in the problem.

p(B| A) = 1.00 This, too, is given in the problem.

So: p(A| B) = 1.00 x ..001

(.0001 x 1) + (.05 x .999)

= .01998 = .02

So p(A| B) is approximately .02