# Probability of gun death

The tragedy in Oregon has reignited the gun debate. Gun control advocates argue that fewer guns mean fewer deaths while gun supporters argue that if citizens were armed then shooters could be stopped through vigilante action. These arguments can be quantified in a simple model of the probability of gun death, $p_d$:

$p_d = p_gp_u(1-p_gp_v) + p_gp_a$

where $p_g$ is the probability of having a gun, $p_u$ is the probability of being a criminal or  mentally unstable enough to become a shooter, $p_v$ is the probability of effective vigilante action, and $p_a$ is the probability of accidental death or suicide.  The probability of being killed by a gun is given by the probability of someone having a gun times the probability that they are unstable enough to use it. This is reduced by the probability of a potential victim having a gun times the probability of acting effectively to stop the shooter. Finally, there is also a probability of dying through an accident.

The first derivative of $p_d$ with respect to $p_g$ is $p_u - 2 p_u p_g p_v + p_a$ and the second derivative is negative. Thus, the minimum of $p_d$ cannot be in the interior $0 < p_g < 1$ and must be at the boundary. Given that $p_d = 0$ when $p_g=0$ and $p_d = p_u(1-p_v) + p_a$ when $p_g = 1$, the absolute minimum is found when no one has a gun. Even if vigilante action was 100% effective, there would still be gun deaths due to accidents. Now, some would argue that zero guns is not possible so we can examine if it is better to have fewer guns or more guns. $p_d$ is maximal at $p_g = (p_u + p_a)/(2p_u p_v)$. Thus, unless $p_v$ is greater than one half then even in the absence of accidents there is no situation where increasing the number of guns makes us safer. The bottom line is that if we want to reduce gun deaths we should either reduce the number of guns or make sure everyone is armed and has military training.

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## 5 thoughts on “Probability of gun death”

1. Sort of interesting excercize –as a start, in quantification. Quantification is an interesting issue—basically formalization or development of a standard vocabulary. I was glancing at 3 papers on this today (all in somewhat questionable free on-line journals). one on the ‘club of rome’ type models of civilization collapse in sept 2012 issue of ‘sustainability’ (part of mdpi which also publishes entropy); one on rwer.wordpress.com (real world economics review) on probabilistic models in neoclassical economics (which seemed a bit confused—they also have an article on guns called ‘follow the gun money’), and one in open journal of philosophy on the use of quantification and formalization. (Most people i sort of know have zero use for quantification–they want to get out the vote, protest for a minimum wage, etc. or set up a timebank (ie labor exchange where people trade hours http://www.dc.timebanks.org — said i’d prefer to also look at a math model to see what critical mass of people and skills need to be involved to make it viable—it doesnt seem functional if u only have 10 people — a sort of market where everyone has something to sell and something they want to buy, but noone wants whats available or offered—‘stable matching theorem’)).

My view of quantification is its useful but has a long time lag—-calculating boltzmann’s constant or collecting rigorous data on cancer causes, etc. is in the long run seen as useful but in the short run is mostly an academic or aesthetic excercize.
(I was looking to see if the gun equation has any obvious interesting properties–i rewrote it using the subscripts as d=g(u+a-guv) which has a sort of epidimiological flavor.)

my view is that actually it is quite likely all the probabilities of being criminal, unstable,an effective vigilant, a suiccide or accident are not fixed or hardwired but dependent on the social environment. (Poverty, malnutrition, bullying, media and propoganda, peer pressure, regulations, etc. may all affect these probabilities). This is why i am somewhat skeptical of formalization. That can just be an industry — i have read many papers in criminology journals many of which seem to be just speculation (which comes with a salary) about why ‘they’ do what they do (some from the view it seems that iits because ‘they’ are defective, otherwise they’d have tenure). . Similarily i know some people who put alot of effort into getting law degrees so they can represent the indigent and nonviolent offenders, but i guess it seems maybe it would be better to figure out a way to create an environment so they dont get into trouble even if in 5 years when you have your degree they can get legal representation. Some go to law school as an alternative to or even after they get in trouble (just as some people join the army because they get a choice of that or a jail sentence)..

politically changing the environment may be almost impossible or so slow the various probabilities are effectively constant. the gun lobby is not going anywhere. .Its interesting also why some people enjoy going to shooting ranges and aquiring guns (there is a whole radio show in dc devoted to this) as opposed to say trying to solve math problems or doing other things. some like james fowler argue on sort of behavioral genetic grounds these kinds of personality types are basically genetically determined, and may actually represent a sort of polymorphism among human psyches—maybe we need hawks and doves, etc. The media certainly seems to thrive off of violent incidents tho that may be due to poverty of imagination (though alot of classic literature from homer and shakespeare to fairy tales actually have violent subtexts).. .

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2. I think an important element of cost analysis is missing in the model: for example the cost of carrying and maintaining a ready gun per person all the time or cost of accidental usage! This cost makes the strategy that “everyone is armed and has military training” is far way from optimal! So the optimal strategy is “[to] reduce the number of guns”. Or sadly keep the current situation as a possible Nash equilibrium!

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3. The zero gun solution is optimal. The armed and trained solution is locally optimal if you have a sufficient number of guns already.

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4. Your missing the point. What makes the most profit for the gun industry, is the only equation the NRA cares about.

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5. Rick G says:

I think the correlations between p_g, p_u, and p_v are probably pretty strong, especially if p_u is about mass-shooter rampage types. Strong enough that I could imagine p_g being above the optimum (and raising p_g reducing p_d by lowering the effectiveness of mass shooting rampages).

However, most gun deaths don’t occur in mass shooting but rather in standoffs between individuals that escalate into violence. And there I’d bet that the probabilities are less likely to be correlated, but more importantly there isn’t even a p_v anymore, it’s just p_d = p_g * p_maximum_escalation, a strictly positive function in p_g!

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