The amazing gun-toting teacher is a fantasy character from the US gun-lobby. The idea is this – when a school shooting occurs the buy-more-guns lobby places the blame on the teacher not being armed. They claim that if the teacher was armed then the teacher could shoot the attacked and thus prevent or reduce casualties. The point of the scenario is to put the blame of the shooting on a lack of guns.
It is fairly transparently nonsense but it occurred to me that it does illustrate an interesting mathematical issue with risk and why people are bad at evaluating it.
First of all we need to put some flesh on the rhetoric. In this NRA fantasy a teacher of a class of children has a gun. I’ll assume the students are too young for even the NRA to suggest that they carry guns to school. Unlike the current USA gun ownership is much greater and open-carrying of weapons has become the norm.
The teacher (T) is in a classroom full of children and the potential attacker (X) enters.
What happens next?
Assuming X is an attacked bent on shooting lots of people, then they would shoot the teacher whom they can reasonably expect to be armed. However, this assumes the teacher does not react first, which is a reasonable assumption as the teacher will not normally expect to be shot by people visiting their classroom.
So not only does the teacher need to be armed but they need to be alert and ready an attack. So let’s play this scenario out again. T is their classroom and X enters. T must now make a quick evaluation: is X a danger? Should T do nothing, or should they draw their gun or should they draw their gun and fire? T needs to make a decision quickly because if X is a shooter then they have an advantage.
On what basis should T make their decision? Is X carrying a gun would in the real world be a sensible criteria but we’ve already assumed we are in a world where the open-carrying of weapons is common place. X carrying a gun is to be expected. T needs to make a snap judgement using everything they know about people. Is this a harmless visitor or is this a killer?
Now as it happens T is a psychological genius. They can tell with 99% accuracy whether somebody is coming into their classroom with murderous intent. What do I mean by that?
Any time a visitor enters T’s classroom they make a judgement- killer or not-killer. We’ll call ‘killer’ a positive and ‘not-killer’ a negative. T’s judgement is either right or wrong and that means we have a neat matrix of outcomes.
1. Judgement: killer. Truth: killer.
2. Judgement: not-killer. Truth: not-killer
3. Judgement: killer. Truth: not-killer
4. Judgement: not-killer. Truth. killer.
3 and 4 have special names – a false positive and a false negative. For 1 out of every 100 not-killer’s T will score a false positive and for 1 out of every 100 killers T will score a false negative. For simplicity I’ve made that symmetrical but T’s proportions for false-positive and false-negative need not be equal.
Now let’s add some numbers. Although any level of school shooting is terrible, they are fairly rare occurrences. Estimates vary because it is unclear exactly what counts. As the NRA fantasy is around a particular kind of scenario we should try and match it closely. I’ll stick with public elementary schools of which there are about 67 thousand in the US. Arguably there are 6 school shootings in elementary schools per year (based on 2013-2014 figures compiled by pro-gun control groups) but this figure includes varying kinds of incidents that don’t necessarily match the scenario we are discussing. I’ll knock it down to 0.5 incidents across the whole of the US in a year. From that we can work out the chance of a teacher T encountering a killer – p(killer) = 0.5/67000=1/134000. If T has a 30 year career, outcome 1 (true-positive) and outcome (4 false-negative) are very unlikely to ever occur.
Outcomes 2 and 3 are going to dominate most outcomes. As T is a good judge of character, the most common outcome will be 2 (true negative) but we can expect one out of 100 outcomes to be type 3, a false-positive. T draws their gun and assess X as a killer and shoots before they do…only X isn’t a killer. 100 outcomes a year will average about one shooting for T alone. I don’t have good numbers but based on violent crime stats on fatal v non-fatal shootings it is about a 1/5 chance of death. So T kills about 1 non-killer every 5 years, or 0.2 a year. Multiply that by, say, 15 teachers per school, that is 3 per year. Multiply that by the number of public elementary schools that comes to about TWO HUNDRED THOUSAND deaths.
OK, but maybe I fudged the numbers. The calculation looks like this 0.01 (false-positive) *100 outcome * 0.2 (fatality) * 15 teachers * 67 thousand schools = 201000. Let’s make T much better at not shooting people. They now have a 0.01% chance of a false positive. 0.0001*100*0.2*15*67000=2010 deaths. If you feel my numbers aren’t right then it is easy to play with them yourself. However a basic fact is this – school shootings are infrequent and traumatic, in theory increased gun ownership could reduce them but the cost in lives would necessarily be greater. Now note, you just need to try out the numbers. You need very few assumptions about shooter’s motives, about the impact of guns on society, the sociology of gun ownership etc or any sort of moral stance on gun control. In addition I have not factored in the additional risks from accidental shootings or any increased likelihood of shootings form gun availability generally. This is a conservative estimate.