Thursday, December 20, 2012

Guns and probability

Let's solve a simple (but unpleasant) probabilistic problem.

  • You are in a room with 19 other people (so 20 in total). One of them turns to be a "freak".
  • If a freak has a gun - they start shooting people until nobody is left.
  • If a "normal person" has a gun, they shoot the freak, and save everybody.
What is the probability of dying in this situation, as a function of gun availability in the country?

Let's assume that the probability of having a gun is the same for everyone in the room, and equals p. Then the probability of the freak being the only armed person in the room is given by the formula d = p*((1-p)^(n-1)), where n=20. If everybody carry their guns openly, and the freak is rational enough; or if "normal people" manage to always kill the freak before the freak kills anybody, this formula will describe the probability of death in this situation. It will obviously go through a maximum, and then decline back to zero:

Let's assume however that "normal citizens" don't shoot the freak until they are 100% sure that this guy is actually a freak. So the freak always succeeds in killing one person, and only then they are stopped by a "militia" member. Then the curve would look slightly differently, because now the probability of death is 1 (for sure) if there's no other armed person in the room, but it is still 1/20 if there are militia members there: d = p*((1-p)^(n-1)*1 + (1-(1-p)^(n-1))*1/n). This is assuming that you are always a "good citizen", and never a freak.

And this formula suddenly makes some sense. It is possible to decrease the probability of shooting sprees by increasing gun ownership. But at some point the shootings will become that frequent, that even if each of them is stopped almost immediately, they will still take a toll. Because in each of them at least one innocent person will be killed. You change the assumptions, and the parameters, and the curve will move around, but the idea will stay the same.

- "Well", - the right-wing person would say, - "but people never turn freaks right in the room; they usually turn mad while at home, and take their time to prepare. What if a freak is 5 times more likely to find a gun than a "normal person", because a freak is actively looking for one?" In this case, indeed, the only way to stop the freaks is to give everybody a gun, because d = (1-(1-p)^5)*((1-p)^(n-1)*1 + (1-(1-p)^(n-1))*1/n).

In reality however all freaks are different, and while there are some who will sale their belongings and prepare for years, most of them will probably kill other people only if given an opportunity. The harder it is to get a gun, the fewer freaks capable of doing so you will find. For example, if on top of "purely opportunistic freaks", as described above, you'll also find 1/2 as many freaks that are twice more persistent in getting a gun, 1/4 as many freaks that are three times as persistent, etc., you'll end up with this curve:

And here basically, in essense, we return to the curve #2. If everybody has a gun - people die all the time. If you reduce gun ownership, rate of murders go down. At some point however you may feel helpless, because if 10% of people walk around with a gun in their pocket, sprees will still happen regularly, and they'll be already quite deadly. That is essentially the situation in any school, or any mall, where the majority of public obey the "gun free zone" laws. However, once you drop the average gun ownership rate below a certain point, gun control becomes the only efficient way to further reduce the casualties.


  1. I was thinking about this. The availability of guns increases the probability of the tragedies we want to avoid. Thanks for putting it in these terms.

  2. It's just common sense. The guns bear a "fixed price", which is proportional to guns availability. In real life it is mostly domestic violence, jealousy murders (crazy ex-boyfriends etc.), accidents, and suicide. But the logic is still about the same. It's highly unlikely that rare glorious cases of brave militia men disarming a bad guy would statistically outweigh the "fixed price".