The gun violence lottery

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A recent New York Times story reported the frustration in trying to determine a “cause” for Stephen Paddock’s October 2017 murder of 58 concertgoers in Las Vegas, and the wounding of hundreds more, all shot using a modified semi-automatic weapon, fired high up from an adjacent hotel. [1] No strong personal motives have emerged, at least significant enough to commit such a well-planned mass murder. And now, an autopsy of his brain has revealed nothing out of the ordinary for a man his age.

Then on February 14, 2017, at least seventeen students were killed in a South Florida high school by a single gunman. In every case of mass shootings, the media pundits quickly jump to the conclusion that there must be “One Cause” for this act of violence, although they disagree as to what that cause is. Is it mental illness? Religious indoctrination? Video games violence? Divine plan?

At the risk of irritating some readers, let me suggest, in the spirit of this blog, that there often is no cause beyond some sort of natural randomness inherent in the base condition of “guns present in unsafe conditions.” Later in this post I will look at gun violence data viewed mathematically from this probabilistic perspective. Sadly, this is also a probability factor in base conditions that we could partially control, but we refuse to do it. Here is the my formula of the minimum base conditions:

Person angry with the world
Available high-capacity weapon
“Random factor X”

The “Random factor X” here is not unlike the stubborn unknown causation for many forms of cancer that I wrote about in a series of earlier posts. At some point human cells just randomly (at least statistically) turn cancerous at eerily-predictable rates. At some low level of probability, guns, if they are present in unsafe places and unsafe hands, will go off. And if they are high-capacity weapons, they will go off with great and horrific consequence.

Lottery mathematics

Americans love lotteries, but most Americans do not understand lottery mathematics. [2] And because they don’t understand lottery mathematics, they haven’t realized that the occurrence of gun violence in America is likely predicted better by the math of lotteries than by any of the other “causes” widely claimed after each horrible new incident. In these tragic events, we watch the “improbable” become the “inevitable.” The “winners” of the gun violence lottery are instead its ill-fated victims.

Lotteries succeed when they satisfy two statistical properties:

  • The mathematical odds of a single “win” are very, very low, perhaps even less than one in ten million tickets sold.
  • The number of entries qualifying for a win or loss is very, very high, often in the tens of millions of tickets sold within a short period of time.

So in the typical lottery, the odds of you winning are almost zero. However, the odds of somebody winning approach 100 percent, thanks to Poisson’s Law of Large Numbers. If you can balance these two probabilities correctly, then you have a successful lottery.

Let’s go back to proposed “causes” for mass gun violence. There are millions of Americans with various degrees and forms of mental illness, but only a very small fraction will act out violently. And, for important reasons, the very small percentage of people with mental illness in most non-US countries who might act violently still don’t commit gun violence. We likewise have millions of religious extremists in this country, but very few act out their extremism in violent ways. Video gamers? Again, tens of millions of them, but only a rare few problem game-related violence cases annually, and almost only in the United States. Anger management problems? Millions of people suffer from this, but only the tiniest percentage of them resort to mass murder.

The statistical correlation of any of these proposed causes is very weak, and their predictive value even weaker. They just do not predict any single case of violence with any reliability. As well, tens of millions of Americans own guns, yet only a small percentage cause harm with those guns, and even then it is much more likely that the gun’s owner, a family member, or an acquaintance is killed with that gun, and most of those by suicide.

To extend the lottery analogy, you can think of each of these conditions as “buying a ticket.” Every gun purchased yields a tiny probability “ticket” that it will be used to kill someone. Every case of untreated mental illness generates a very low-probability “ticket” that may result in a factor of violence down the road. An obsession with fantasy violence is a frequent side of adolescence, but only a small handful are holding an ill-fated “ticket” of tendency to gun violence. Every multi-round ammo clip sold bears a tiny, but measurable, probability that it will be the “ticket” that turns a single shooting into a even more horrible mass shooting.

So, like the lottery, the odds of any one person tragically “winning” this lottery, either as the perpetrator or the victim of random gun violence, are very, very small. Yet because the Law of Large Numbers takes over, there will be more mass killings in the future, because the number of “entries” among these various “cause lotteries,” especially the count of unsecured guns, is such a massively-large number in the United States right now. One of those “tickets” is likely to be a winner, as we shall see, literally as you read this today.

The statistical correlation to gun violence is not strong for any “one cause,” and even those weak correlations are not necessarily causation. Instead, the primary statistical difference in the United States as compared to the rest of the developed world is found in the vast difference in number of “tickets” sold. When the number of “tickets” increases by a factor of ten, or even one hundred, as it is here compared to many other countries, so do the incidents of gun violence increase by the same factor.

Poisson randomness and gun violence

The Gun Violence Archive attempts to tally and document each reported case of non-suicide gun violence in the U. S. They define a “mass shooting” as an event in which four or more people are injured or killed by a gun. In 2017, they documented 346 of these incidents. This was an average of 0.95 incidents per day, or almost one per day.

Why do I insist that some sort of “natural randomness” is present in gun violence? In a prior post, I discussed the Poisson probability distribution (pronounced “pwa sawn”), which is “nature’s distribution” of how often seemingly-random, but low-probability events happen over time, such as normal cell replication, or cancerous cell replication, or radioactive atomic decay. Using the Gun Violence Archive’s data, and using the 2017 average number of mass shooting per day (0.95) as the “lambda” (mean) of the Poisson distribution, I calculated the following comparison of the actual number of mass shootings reported on any given day in 2017 versus what “natural randomness,” at this rate, would project.

2017 Mass Shootings

What is this graph showing? The Poisson formula predicts how many events we can expect to occur if they are independent from each other, but occur at a particular rate, 0.95 per day in this case. The Poisson prediction for the number of days in one year with zero mass shootings calculates as 141, with 134 days predicted to have one mass shooting, 64 days having two shootings, 20 days having three shootings, and going down from there. [3]

In this observation of 2017 data, the Poisson distribution underestimates by a bit the number of “zero shooting” days and overestimates the number of “one shooting” days. Interestingly though, the total of predicted “zero-shooting” plus “one-shooting” days comes out to 175, and that happens to be the exact number of “zero-shooting” plus “one-shooting” days actually occurring during calendar 2017. The higher-incident days are even closer to their predicted values.

The comparable data for 2016, which had a slightly-higher average of shootings per day (1.05), demonstrates a similar relationship. It is important to note that this is not proof of “Poisson randomness,” but it does suggest that we do have a phenomenon in U.S. mass shootings where we can tell, within an eerie accuracy, “How many?” mass murder incidents we will have today. We just can’t get a handle on “Who, specifically?” But if this is indeed probabilistically random, that level of prediction may be impossible, just as we can accurately predict how many people will be diagnosed with a particular form of cancer this year, but cannot predict who.

If this graph doesn’t convince you, then let’s go to bigger mass shooting incidents, those that have ten or more victims. In 2017, there were eleven of these incidents, almost one per month. But importantly, pure Poisson randomness would predict that there would be several entire months when zero 10+ shootings occur (approximately 5), as well a predicted number of months to experience one shooting (approximately 4), two shootings, three shootings, etc. Let look at the actual versus predicted for monthly 10+ shootings in 2017. [3]

2017 Mass Shootings 10-plus

In reality, these rates of approximately one mass shooting per day, or one “10+” mass shooting per month, are likely the convergence of dozens of potential causes, each with a very tiny probability of occurring. In one sense, then, lots of causes.  But in another sense, because there are so many, each with such low odds by itself, there is effectively “no cause” other than, “lots and lots of tickets” have been sold, just as some cancers seem to be just a function of “lots and lots” of cells replicating, and one goes bad.

So, now what?

Every lottery director knows that it is possible to control both the frequency and size of lottery wins. It is in the math of the Law of Large Numbers and Poisson probability distributions. In practice, this means you try to control the number of tickets sold and their price.

When it comes to gun violence, we know this Poisson “lambda” (mean) number, not just nationwide, but for each state and demographic group. And we know it for comparable countries with different variations on gun laws. Regardless of “cause,” then, we can both measure the current numbers and quantify the effect of any attempted changes to laws and policy. We can’t eliminate gun violence, but we can lower the lambda, and that means real lives saved.

It is possible to greatly reduce the probability of a cruel gun violence lottery “win,” as well as lowering the size of the obscene “big payout,” the number of people killed in a mass shooting incident. We will never get this down to zero, but we can reduce the impacts a factor of ten or more. How do we know that? Because other countries do it. Because they want to do it. We just need the courage to stop selling so many damned “tickets” and handing them out like candy to the wrong people.

The choice is yours…

Note than an update to these numbers halfway through calendar 2018 has been posted.


  1. Fink, Sheri. “Las Vegas Gunman’s Brain Exam Only Deepens Mystery of His Actions.” The New York Times, 9 Feb. 2018.
  2. See my longer post on lottery math and the Law of Large Numbers.
  3. Here is the data for 4+ daily mass shootings, followed by 10+ monthly shootings, from the Gun Violence Archive.

Poisson probabilities

Poisson shootings 10-plus
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8 thoughts on “The gun violence lottery

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