I wrote: [Snip application of Bayes's Theorem to facial recognition in airports looking for known terrorism suspects]
And the point is, this sort of fallacy occurs _all the time_ when people talk about probilities and rates of success for infrequent events and large amounts of data.
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Once again, classic base rate fallacy. The 'failure rate of 0.008%' figure is totally wrong.
There are several _very_ common statistical fallacies people commit in situations involving low probabilities. Poor understanding of probabilities and statistics is increasingly becoming a huge problem -- in criminal jury trials, setting of government policy, and elsewhere. o Base-Rate Fallacy: irrationally disregarding _completely_ general likelihood information ('base rate' information) you have about rarity of something, because you've focussed on specific details of a case. (The better alternative is to weigh both types of information appropriately.) o Prosecutor's Fallacy: confusing the likelihood of a suspect satisfying a description with the likelihood of any person satisfying the description being guilty -- i.e., confuses the likelihood of seeing some evidence at all with the likelihood of that evidence indicating guily. o Defendant's Fallacy: confusing extreme rarity of a trait with very low likelihood of someone having that trait being guilty. The Wikipedia explanations of all three are _utterly_ wretched. I'm a mathematician, and *I* find those write-ups confusing and useless. For some reason, this really bothers me. I guess I like clarity. This page, unlike the execrable Wikipedia one, explains Prosecutor's Fallacy really well: http://www.conceptstew.co.uk/pages/prosecutors_fallacy.html This one is pretty good: http://www.agenarisk.com/resources/probability_puzzles/prosecutor.shtml These pages do a so-so job of explaining Base-Rate Fallacy: https://www.logicallyfallacious.com/tools/lp/Bo/LogicalFallacies/55/Base_Rat... http://www.fallacyfiles.org/baserate.html This one explains Defendant's Fallacy pretty well: http://www.agenarisk.com/resources/probability_puzzles/defendant.shtml All three fallacies go away if you _apply_ Bayes's Theorem to proability problems, but hardly anyone will do that (without goading, anyway), and it'd be better if folks could better master probability intuitively. I like the conceptstew.co.uk page's rundown on Prosecutor's Fallacy (using a hypothetical criminal case) so much, I'm going to summarise it below: A purse-snatcher in London absconded with quite a lot of cash, but the victim gave a detailed description of the thief that included several distinctive physical traits. A suspect got picked up the next day who matched all those traits. He was arraigned. There was no other physical evidence. At trial, the Crown argued population probabilities, quizzing as expert witness a government statistician: 'Being male is 0.51 likely. Being 2 metres tall is 0.025. Being between 20 and 30 years old is 0.25. Being red-headed is 0.037. Having a pronounced limp is 0.017.' Likelihood of all these independent traits at once is 0.51 x 0.025 x 0.25 x 0.037 x 0.017 = 0.000002 -- or about one in half a million. 'The chance of any random individual sharing all these characteristics is vanishingly small - only 0.000002. The prisoner has them all.' So, he argued, the chance of him being innocent is infinitesimal, and he should be convicted. Defence counsel took the stand, and cross-examined the expert witness: 'What is the population of London?' He looked startled, but replied 'I think it is about 10 million.' 'So based on your statistics, how many people in London have this set of characteristics?' He blustered a bit, but was forced to admit that there should be 20. 'Given that your evidence is based solely on a description and that you have admitted that there are 20 people in London who fit this description, this must mean that the probability of my client's guilt is very small, only only 1 in 20. Or to put it another way, the chance of his innocence is 19 in 20, not 1 in half-a-million.' Page concludes: The expert witness [during direct testimony] confused two things: o the probability of an individual matching a description, and o the probability of an individual who _does_ match that description being guilty They are not the same! It is easier to see the fallacy as soon as the probability of 0.000002 is turned into numbers of real people: When you bear in mind that the population of possible suspects is 10 million, 1 in half a million easily translates into 20 possible suspects - the accused is only one of this group, and, if we are to be convinced of his guilt with no other evidence, we would want to know that the other 19 had been excluded. And that is without even considering people who might have come up to London for the day!