### Making Decisions About Suspected Suicide Bombers

**The shooting of the Brazilian in London has generated little rational discussion in the blogosphere. The nasty reality is that the suicide bomber forces a rational police officer to shoot dead a suspect if he is only 10% confident that an individual is about to detonate. People who are taking an anti-police line (including the otherwise estimable Bruce Schneier) would do better proposing really effective non-lethal options for the police.**

There's a lot of theory on the problem of choosing between unpleasant alternatives, and the one I use is based on the work of this man.

*Thomas Bayes, one of the leading mathematical lights in computing today, differs from most of his colleagues: He has argued that the existence of God can be derived from equations. His most important paper was published by someone else. And he's been dead for 241 years.*

Here's how it works.

Imagine you are a London cop facing a suspected suicide bomber in a crowded railway carriage. He might pull the trigger at any moment, so you have to make a split-second decision as to whether or not to shoot him.

Your estimate that the guy is a suicide bomber is P, a number between zero (which would be certainty that he was innocent), and 1 (certainty that he was a bomber). It follows that your estimated probability that the guy is

**not**a suicide bomber is (1-P).

You quickly review the

**costs**of your 2 options:

- if you don't shoot him and he detonates, you and a bunch of innocents will be maimed or killed.

- if you do shoot him, and he's innocent, he for sure is dead.

(Incidentally, the references to a "shoot-to-kill-policy" are idiotic - the first thing you learn in weapons training is that you must always assume that your shot will kill.)

Back to statistics. We can calculate the costs of the 2 outcomes our police officer faces:

- in the four 7/7/ bombings, 52 innocents died - each suicide bomber killed 13 people.

- the cost of shooting an innocent is obviously one dead.

So now we can calculate the

**probabilistic costs**which combine probability with cost:

- if you do not shoot the guy, he has probability P of detonating, so the probabilistic cost is 13 dead * P

- if you shoot him and he's innocent (probability (1-P), the probabilistic cost is 1 dead * (1-P)

A Bayesian decision is to shoot if the probabilistic cost of shooting a bomber (13*P) is greater than probabilistic cost of shooting an innocent (1*(1-P)). When the two are equal (13*P = 1*(1-P)), the decision is balanced

Solving for P gives a balanced decision value for P of just

**7%**, and allowing a margin would set the level at

**10%**.

**In summary, you save the most lives if you shoot a suspect when you have a better than 1 in 10 belief that he is a bomber. Which means 9 innocents get shot for every 1 suicide bomber! But killing the latter saves 13 innocents, so you're ahead.**

(For statistician readers: I've run other models with added costs - injury, economic, intelligence, and also looked at multiple correlated outcomes. Still comes out around 10%).

Being expected to kill on such low probabilities must be an impossible burden for the police officers who are making and executing these decisions & I bet they use much higher probabilities, even though that means more innocents die. So in a later post I'll look at ways to reduce this pressure and loss of innocent lives using non-lethal options.

**UPDATE**

A warm welcome to visitors from Tim Worstall and the BBC Magazine. As you'll see from the above, this blog tries to combine hard data with basic scienctific disciplines to make sense of the world.

Don't be put off by the maths - we only use the intuitively obvious stuff - statistical inference, threat assessment, queuing theory etc. For example, take a look at my take on the London police and US TSA policies on not selectively searching people from the same racial background as previous terrorists. You might agree with these policies, but the numbers show that there's a cost.

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