Bruce Schneier pointed out this BBC News article about the base rate fallacy
. In that article, they consider a Terrorist Detector
that is right 90% of the time. Given that the test signals a positive for someone, how sure are you the person in question is a terrorist?
The gist of the math
Well, the math in that article is confusing. They say the answer is about 0.3%. But, they later do a diagram that makes it look like 1 in 100,000 to me. Further, they tacitly assume that while 10% of the time the test will identify a non-terrorist as a terrorist, it will never miss a real terrorist.
Regardless, the goal of the article is to show Bayes’s Theorem in action for something that has a low prior probability. In layman’s terms, if something is really, really unlikely, then you are very bad off if your test is only really good. It’s really unlikely that any given person is a terrorist. As such, if your test thinks every tenth person is a terrorist, your test is almost never right when it thinks it has found a terrorist.
In the article, they ask for help describing this fallacy in ways that will drive the point home to people on a gut level. I can kind of see how one could miss the idea in cancer screening. But, with terrorism?
The real world
Remember the last time you were on a plane? [Okay, I do… it was last Thursday.] You were sitting there with about a hundred fellow passengers. How many of them do you think were terrorists? My guess is that you were pretty sure (even before the flight took off) that zero of them were terrorists. How many of them bombed your flight? I’m pretty sure that number was zero (and not just because you wouldn’t be reading this otherwise). How many of you did the test think were terrorists? About ten.
What proportion of hundred-passenger flights would the test think are terrorist-free? Fewer than one in every 35,000 flights. Buy a lottery ticket if you’re on a flight where this (hypothetical) test thinks there are no terrorists because it just may be your lucky day.
To do the full analysis, you have to consider how many of those 35,000 flights really do have terrorists on them. But, it should be blatantly obvious to everyone that the answer is way, way under 34,999.
What is a terrorist, anyway?
This brings up another question that I always have when people start talking about detecting terrorists
or the number of terrorists in the U.S.
, etc. When is someone a terrorist? If Khalid Sheikh Mohammed were traveling to Hawaii for some R&R, is he a terrorist? would the test detect him? should he be allowed on the plane?
Detecting terrorists is often talked about like detecting diseases and such. But, really, it is about detecting intentions. Intentions are far more evanescent than microbes.
Actually, on a second reading, it looks like the article may only be thinking that the test will correctly identify 90% of the terrorists, too. In their sample math in that section, they assume there’s only one real terrorist and the test finds that terrorist. In a later section though, they seem to assume there are 300 terrorists and the test would find all of them rather than just 270 of them.