Editing Base rate fallacy (section)

===Example 3: Terrorist identification===
In a city of 1 million inhabitants, let there be 100 terrorists and 999,900 non-terrorists. To simplify the example, it is assumed that all people present 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%:
* The false negative rate: If the camera scans a terrorist, a bell will ring 99% of the time, and it will fail to ring 1% of the time.
* The false positive rate: If the camera scans 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. Someone making the base rate fallacy would infer that there is a 99% probability 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 probability of a terrorist is 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' (P(¬B | T), or the probability that the bell fails to ring given the inhabitant is a terrorist) and the 'number of non-terrorists per 100 bells' (P(¬T | B), or the probability that the inhabitant is a non-terrorist given the bell rings) are unrelated quantities; one is not necessarily equal—or even close—to the other. 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 first 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. The probability that a person triggering the alarm actually is a terrorist is only about 99 in 10,098, which is less than 1% and very, very far below the initial guess of 99%.

The base rate fallacy is so misleading in this example because there are many more non-terrorists than terrorists, and the number of false positives (non-terrorists scanned as terrorists) is so much larger than the true positives (terrorists scanned as terrorists).

Multiple practitioners have argued that as the base rate of terrorism is extremely low, using [[data mining]] and predictive algorithms to identify terrorists cannot feasibly work due to the false positive paradox.<ref name=":0">{{Cite journal |last=Munk |first=Timme Bisgaard |date=1 September 2017 |title=100,000 false positives for every real terrorist: Why anti-terror algorithms don't work |url=https://firstmonday.org/ojs/index.php/fm/article/view/7126 |journal=First Monday |volume=22 |issue=9 |doi=10.5210/fm.v22i9.7126 |doi-access=free}}</ref><ref name=":1">{{Cite magazine |last=Schneier |first=Bruce |author-link=Bruce Schneier |title=Why Data Mining Won't Stop Terror |url=https://www.wired.com/2006/03/why-data-mining-wont-stop-terror-2/ |access-date=2022-08-30 |magazine=Wired |language=en-US |issn=1059-1028}}</ref><ref name=":2">{{Cite web |last1=Jonas |first1=Jeff |last2=Harper |first2=Jim |date=2006-12-11 |title=Effective Counterterrorism and the Limited Role of Predictive Data Mining |url=https://www.cato.org/policy-analysis/effective-counterterrorism-limited-role-predictive-data-mining# |access-date=2022-08-30 |website=[[Cato Institute]]}}</ref><ref name=":3">{{Cite journal |last=Sageman |first=Marc |author-link=Marc Sageman |date=2021-02-17 |title=The Implication of Terrorism's Extremely Low Base Rate |url=https://doi.org/10.1080/09546553.2021.1880226 |journal=Terrorism and Political Violence |volume=33 |issue=2 |pages=302–311 |doi=10.1080/09546553.2021.1880226 |issn=0954-6553 |s2cid=232341781}}</ref> Estimates of the number of false positives for each accurate result vary from over ten thousand<ref name=":3" /> to one billion;<ref name=":1" /> consequently, investigating each lead would be cost- and time-prohibitive.<ref name=":0" /><ref name=":2" /> The level of accuracy required to make these models viable is likely unachievable. Foremost, the low base rate of terrorism also means there is a lack of data with which to make an accurate algorithm.<ref name=":2" /> Further, in the context of detecting terrorism false negatives are highly undesirable and thus must be minimised as much as possible; however, this requires [[Sensitivity and specificity|increasing sensitivity at the cost of specificity]], increasing false positives.<ref name=":3" /> It is also questionable whether the use of such models by law enforcement would meet the requisite [[Burden of proof (law)|burden of proof]] given that over 99% of results would be false positives.<ref name=":3" />