If a test to detect a disease whose prevalence is 1 out of 1,000 has a false positive rate of 5 percent, what is the chance that a person found to have a positive result actually has the disease?

About 2%

Step-by-step explanation:

Let call A the case where a person has the disease, and B the case where the person got a positive in the test. The chance that a person found to have a positive result actually has the disease (P (A|B)) can be found using Bayes' theorem:

P (A|B) = P (B|A) * P (A)

P (B)

P (B|A) is the probability of obtaining a positive result in the test in the case that we actually know that we have the disease. This value would be 1. P (A) is the probability of having the disease, and it is said by the problem: 1/1000. P (B) is the probability of obtaining a positive test. Be careful, this positive test can be a real positive test or a false positive test. We just care about obtaing a positive. This would be the probaility of obtaining a real positive plus the probability of obtaining a false positive.

The probability of obtaining a real positive is the probability of having the disease and obtaining a positive. Notice we already established how to calculate this value (P (B|A) * P (A))

The probability of obtaining a false positive is the probability of not having the disease (999/1000) and obtaining a positive (5%).

Then:

P (A|B) = P (B|A) * P (A) = 1 * 1/1000 = 0.1962 or about 2%

P (B) 1*1/1000 + 999/1000 * 0.05