False Positive Paradox
Imagine there is a virus that is always asymptomatic in males. 1% of males are infected with it.
There’s a rapid test that never produces false negatives (i.e., it never misses an infection), but it has a 5% false positive rate.
Phil takes the test. It comes back positive. What are the chances he is infected?
Answer
A common answer is 95%, but it’s actually only 16.8%.
This is surprising to many people, but remember that only 1% of males have the disease.
In a group of 10,000 males, we can expect that 100 will have the disease and 9,900 will not.
Imagine we test everyone in the group.
All 100 people who have the disease will test positive since there are no false negatives.
Of the 9,900 who don’t have the disease, 9,405 will test negative, and 495 will show false positives.
This means there are 595 positives: 100 of them are true positives and 495 are false positives.
100 true positives ÷ 595 total positives = 16.8% of the people who test positive are true positives
The False Positive Paradox teaches us that when the prevalence of a disease is low, widespread testing of asymptomatic people leads to a high number of false positives.
