rfbooth.com :: thoughts :: don't shoot, I'm only the piano player

Shortly after some suicide bombers blew themselves, and others, up on the Tube, police shot a Brazilian non-terrorist in the head seven times. Both these events might reasonably make people nervous. The rather Orwellian term “shoot to kill to protect” was used to describe the policy, and the chief of the Met said that the policy would remain, and that more innocent people could be killed while fighting terrorism.

This is exactly the sort of situation - lots of people, very few of whom are actually the ones you want - where simple probabilistic analysis throws up very surprising results. Let's make a model of the situation and see what happens. (Note for the hard of thinking: I do not claim that this model is actually representative of the policy adopted by the police, or that it applies to the Menezes incident). What we're trying to do is work out how likely it is, under the model, that a person the police have killed will turn out to be a suicide bomber.

Here's the model. We assume that the police look at some (but of course not all) people as they go into the Tube. The actual proportion doesn't matter. We assume that anybody they see who is a suicide bomber will be shot (generous to the police, I think it's fair to say; bombers probably try quite hard not to look like bombers). We assume that only 1 in a million non-bombers looked at will be misidentified and shot. All of these numbers are, of course, pure invention.

Let's assume that there are 2.6 million passengers per day (roughly correct), and that there are 20 suicide bombers a year (more than the total number ever, so far, but we can assume escalation). So there are 2.6×365=949 people per year who would be wrongly shot if they were looked at, and 20 who would be rightly shot. So, the probability we're after is 20/969, or about 2%. In other words, under this model, we'd kill more than 47 innocent people for every terrorist we killed.

Putting that in perspective, if your objective is to improve the ratio between suicide bombers killed and commuters killed, the actual bombings killed 4 bombers and 52 victims: 13 to 1. Thus, you're more than three times better off letting people you think are bombers blow themselves up than shooting them, in terms of innocent deaths. Given the rather generous-looking assumptions in our model, you may find this surprising.

If we want to keep the same model and get down to a point where, when you think it's a terrorist, you have a 50:50 chance of being right, you need to assume that the police will wrongly identify only around one in 47.5 million commuters as a terrorist. Of course this may be so. If we only want that thirteen-to-one ratio that the bombers managed in early July, we can settle for about 3.7 million to one; this is the point at which shooting a suspected bomber doesn't actually perform worse than waiting to see if he explodes.

Of course the model or its parameters may be wildly wrong, but it does neatly illustrate how badly that nice clean one-in-a-million does for you in practice. We do not, mathematically, attempt to say what sort of ratio of dead innocents to dead terrorists is acceptable; just that you need to be a great deal better at discriminating between the two categories before they're dead than you would think.