A test of a disease presents a rate of 5% false positives. The disease strikes 1/1000 of the population. People are tested at random, regardless of whether they are suspected of having the disease. A patient’s test is positive.
What is the probability of the patient being stricken with the disease?
Most doctors answer 95%. Less than one in 5 professionals answered the question correctly.
To solve this problem, break the problem down into steps using tables. The 2×2 table will be one that is used in appraisal of diagnostic tests.

 
Fill in the numbers

1. Total number of people with the disease is 1
2. 999 people do not have the disease
3. Therefore, True Positive = 1 and False Negative = 0
4. False positives = 0.05×999 = 49.95 = 50 (5% false positives)
5. True Negative = 1 – False Positive = 949

 
To determine the probability of the patient having the disease, calculate the positive predictive value (PPV) which is different from sensitivity or specificity.
PPV = TruePositive / TruePositive + FalsePositive = 1 / 51 = 2%
Therefore**, there is a 2% probability that the patient has a disease.**
The probability so low, even though the test has only 5% false positives, because of the low incidence rate of the disease. In rare outcomes, the predictive value of any instrument drops.

TALEB'S DESCRIPTION OF SOLUTION

I will simplify the answer using the frequency approach. Assume no false negatives. Consider that out of 1000 patients who are administered the test, one will be expected to be afflicted with the disease.
Out of a population of the remaining 999 healthy patients, the test will identify about 50 with the disease (it is 95%) accurate.
The correct answer should be that the probability of being afflicted with disease for someone selected at random who presented a positive test is the following ratio: Number of afflicted persons divided by number of true positives and false positives. Here, 1 in 51 (2%).