False Negative, False Positive and the Paradox

First a bit about the terms False Positive & False Negative. There terms are associated with the nature of error in the results churned out by a system trying to answer an unknown problem, based on a (limited) set of given/ input data points. After analysing the data, the system is expected to come up with a Yes (it is Positive) or a No (it is Negative) type answer. There is invariably some error in the answer due to noisy data, wrong assumptions, calculation mistakes, unanticipated cases, mechanical errors, surges, etc.

A False Positive is when the system says the answer is Positive, but the answer is actually wrong. An example would be a sensitive car's burglar alarm system that starts to beep due to heavy lightning & thunder on a rainy day. The alarm at this stage is indicating a positive hit (i.e. a burglary), which is not really happening.

On the other hand, a False Negative is when the system answers in a Negative, where the answer should have been a Positive. False negatives happen often with first level medical tests and scans which are unable to detect the cause of pain or discomfort. The test report of "Nothing Abnormal Detected" at this stage is often a False Negative, as revealed by more detailed tests performed later.

The False Positive Paradox is an interesting phenomenon where the likelihood of a False Positive shoots up significantly (& sometimes beyond the actual positive) when the actual rate of occurrence of a condition within a given sample group is very low. The results are thanks to basic likelihood calculations as shown below.

Let's say in a group of size 1,000,000 (1 Mn.), 10% are doctors. Let's say there's a system wherein you feed in a person's Unique ID (UID) and it tells you if the person is a doctor or not. The system has a 0.01% chance of incorrectly reporting a person who is not a doctor to be a doctor (a False Positive).

Now, let's work out our confidence levels of the results given out by the system.


On the other hand if just 0.01% of people in the group are actually doctors (while the rest of the info. remains same) the confidence level works out to be quite different.


This clearly shows that the likelihood of the answer being a False Positive has shot up from much under 1% to as much as 50%, when the occurrence of a condition (number of doctors) within a given population dropped from 10%  (i.e. 100,000) to a low value of 0.1% (i.e. 1,000).