Base Rate Fallacy - Example

Example

In a city of 1 million inhabitants there are 100 terrorists and 999,900 non-terrorists. To simplify the example, it is assumed that the only people in the city are inhabitants. Thus, the base rate probability of a randomly selected inhabitant of the city being a terrorist is 0.0001, and the base rate probability of that same inhabitant being a non-terrorist is 0.9999. In an attempt to catch the terrorists, the city installs an alarm system with a surveillance camera and automatic facial recognition software. The software has two failure rates of 1%:

1. The false negative rate: If the camera sees a terrorist, a bell will ring 99% of the time, and it will fail to ring 1% of the time.
2. The false positive rate: If the camera sees a non-terrorist, a bell will not ring 99% of the time, but it will ring 1% of the time.

Suppose now that an inhabitant triggers the alarm. What is the chance that the person is a terrorist? In other words, what is P(T|B), the probability that a terrorist has been detected given the ringing of the bell? Someone making the 'base rate fallacy' would infer that there is a 99% chance that the detected person is a terrorist. Although the inference seems to make sense, it is actually bad reasoning, and a calculation below will show that the chances they are a terrorist are actually near 1%, not near 99%.

The fallacy arises from confusing the natures of two different failure rates. The 'number of non-bells per 100 terrorists' and the 'number of non-terrorists per 100 bells' are unrelated quantities. One does not necessarily equal the other, and they don't even have to be almost equal. To show this, consider what happens if an identical alarm system were set up in a second city with no terrorists at all. As in the first city, the alarm sounds for 1 out of every 100 non-terrorist inhabitants detected, but unlike in the first city, the alarm never sounds for a terrorist. Therefore 100% of all occasions of the alarm sounding are for non-terrorists, but a false negative rate cannot even be calculated. The 'number of non-terrorists per 100 bells' in that city is 100, yet P(T|B) = 0%. There is zero chance that a terrorist has been detected given the ringing of the bell.

Imagine that the city's entire population of one million people pass in front of the camera. About 99 of the 100 terrorists will trigger the alarm—-and so will about 9,999 of the 999,900 non-terrorists. Therefore, about 10,098 people will trigger the alarm, among which about 99 will be terrorists. So the probability that a person triggering the alarm is actually a terrorist is only about 99 in 10,098, which is less than 1%, and very very far below our initial guess of 99%.

The base rate fallacy is so misleading in this example because there are many more non-terrorists than terrorists. If, instead, the city had about as many terrorists as non-terrorists, and the false-positive rate and the false-negative rate were nearly equal, then the probability of misidentification would be about the same as the false-positive rate of the device. These special conditions hold sometimes: as for instance, about half the women undergoing a pregnancy test are actually pregnant, and some pregnancy tests give about the same rates of false positives and of false negatives. In this case, the rate of false positives per positive test will be nearly equal to the rate of false positives per nonpregnant woman. This is why it is very easy to fall into this fallacy: by coincidence it gives the correct answer in many common situations.

In many real-world situations, though, particularly problems like detecting criminals in a largely law-abiding population, the small proportion of targets in the large population makes the base rate fallacy very applicable. Even a very low false-positive rate will result in so many false alarms as to make such a system useless in practice.