Why do cops, and home and shop owners, empty their guns when shooting at bad guys? Another way to ask this is our title: How Many Times Must You Shoot To Be 95% Sure Of Hitting Your Target At Least Once?

This video explains:

(By the way, I have another new video, but no post, on The Sports Illustrated Curse Explained!)

If you don’t like video (mostly me neither), here’s the explanation.

All probability is conditional on the premises assumed. Which means there is no single unique answer to our question. We can only answer by making different assumptions.

One assumption is the kind of weapon used. Obviously, if you’re launching a nuke at your opponent, the chance you hit him in one shot is high—but perhaps not utterly certain, if we entertain as a premise the possibility of flawed electronics. Our rule in this and in *all* questions of uncertainty is: change the assumptions, change the probability: nothing *has* a probability. Nothing.

If you’re using a 12-gauge with 00 buckshot the chance of hitting is different than if you’re using a pistol loaded with 9 mm ammo. But even that is intolerably vague, because we said nothing about the situations, and the situations you’d use a shotgun do not overlap completely with those you’d use a pistol. Most do not walk down the streets with shotguns (these days, but ask me about pre-madness days up north Michigan in the 70s!), but many do carry pistols.

So we need to firm up our premises if we’re to have any hope of answering our question.

One way to do this is looking at any data on the subject we can find, like cop shootings. Finding that data is not so easy! One example is from the paper “Hitting (or missing) the mark: An examination of police shooting accuracy in officer-involved shooting incidents” by Donner and Popovich in *Policing*.

They track the type of scenarios we’re interested in. You’re confronted by a bad guy and have to shoot. The bad guy is not sitting still waiting to take the shots. He is at least moving, if not running at you, and may himself be shooting or attacking you.

That’s still loose, since it’s possible to envision a host of sub-scenarios in this large one. Like one in which the bad guy is shooting at you from inside a building 100 yards away versus one in which you are being mugged. Or one in which you are being knifed by a youthful teen juvenile scholar and aspiring rap artist.

So we have to speak “on average” and of the types of scenarios in which the cops studied in the paper faced.

These were cops in Dallas in policing situations from 2003 to 2017. According to their data, cops took 354 shots in those fourteen years, hitting with 123 of them, for a 35% hit rate. This hit rate varied from year to year, from a low of 0% to a high of 100%.

But this is really the wrong number because it averages across all separate incidents. What we really want is the chance a *single* shot hits *per* incident; “inside” each incident, I mean. We don’t have that, so we’ll have to guess.

I looked at four skill levels. Shooters who hit with 40% of the shots they take, then 30%, 20% and 10%. In hostile situations, I mean. In which the target is not waiting for you to hit it and may be shooting or knifing back. I accept all corrections and opinions on this, but I think 40% *per shot* represents at least very good skill (yes, *some* are better, but most are not). If I’m wrong, I’ll give you the tool you need below to recalculate based on whatever number you think is reasonable.

Take the best shooter, who hits 40% of his shots. If he shoots once, he has, by our assumption, a 40% chance of hitting. If he takes two shots he has, I hope it is obvious, a greater chance of at least one shot landing. If he takes three, his chance of at least one is greater still. And so on.

What we want is the number (N) of shots he has to take to be 95% sure he hits at least one time.

We can get that through this formula:

0.95 < 1 – (1 – p)^{N},

where p is the chance any individual shot hits, 95% is our limit, and by solving for N. Which is

N > log(1 – 0.95)/ log(1 – p).

(Take the natural log, not log 10.)

For instance, with p = 0.4 (our 40%), N > 5.86. Since we can only shoot whole numbers, a person who hits 40% of the time has to shoot *at least 6 times* to be 95% sure of hitting at least once.

That’s a lot of shots! Empties most revolvers. But it already explains why people have to shoot so much in these situations. They must, because they need at least a 95% of hitting their target. Remember, this is life or death.

Let’s look at the range of abilities, and a range of certainties, in this plot.

The purpleish line is the highest skill level of 40 (40% of shots hit). As you can see, the probability of at least one hit in one shot is, *ta da!*, 0.4.

The 95% certainly line is dashed red, and the 99% is dotted green.

The 40-skill line crosses the 95% line at just over 5 shots, as we calculated. So we have to shoot 6 times to be 95% sure of hitting at least once. And that same line crosses the *99% line* at 10 shots. So we have to shoot 10 times to be 99% sure of hitting at least once.

You can work out the rest yourself. It’s clear that the person with 10% skill (which may well be me!) really needs to practice more. To get to 95% he needs to shoot 28 times! That’s more than most magazines, in pistols anyway. That means reloading, which makes dire situations even worse.

All this is only *hitting* at least once. It says nothing about stopping, killing, or even discouraging your target. To do that, more shots are usually needed, unless you’re in a cartoon. And then you only need one, or even less than one, shot to kill each bad guy. Lots of luck with that in real life.

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Categories: Statistics

This scenario would be a good real-life example for class discussions on binomial experiments and hence statistical modeling.

The calculation requires the assumption that the probability p of scoring remains the same for each shot. If the target is on the run, the probability probably wouldn’t be the same.

Standard silhouette target, stationary, daylight, fair weather to light rain:

With a pistol, up to 25 meters: Once.

With a rifle, up to 300 meters: Once.

With a shotgun, up to 30 meters: Undetermined, having never successfully hit a target with one.

A better example would involve the P(given conditions x|missile hits within y meters) because the firing and targeting processes stay largely constant – ignoring effects like condensation on the cone during liftoff etc.

I have not read Donner et al, but the first reality of shooting at attackers is that you typically do not know whether any shot hits until it’s over, so you keep shooting because (a) you’re over-stimulated; and (b) the other guy (or dog or…) keeps moving – even if he’s about to die from shot one.

The second reality is more interesting in some ways: second shots fired quickly from auto or semi-auto weapons rarely hit the target. Studies using marked bullets in burst-of-three weapons invariably show the 2nd and 3rd wasted unless the weapon is bench mounted (or sand bagged or..otherwise firmly held) – and, placing single shot-at-a-time riflemen with semi-auto weapons under time stress (must fire 3 times in 2.5 secs) has the same effect: #2 almost always misses completely – but people using weapons requiring manual intervention to recharge -e.g. bolt actions – while slower, generally do much better.

JH,

Yes, indeed, shooting into your target depends on a host of circumstances that you try and maximize in your favor.

McChuck,

This is why I like shotguns.

Paul,

Yes, excellent point about not knowing whether you hit.

All of this may be true, but hitting the guy isn’t the only objective. There is a second and equally important objective, which is not to hit innocent bystanders. This is why shooting over 1000 rounds in the middle of a traffic jam at rush hour, something that actually happened in Florida, should result in prison time.

Any non-cop who shoots at a bad guy and misses and hits an innocent person is going to jail even if the shots were justified. But not the cops. The cops shot a hostage victim deliberately and no charges were filed. A bad guy had a logging truck driver at gun point and the police decided it was a safety risk that justified shooting the hostage. They didn’t shoot the bad guy. This just happened a few months ago.

I suppose this is why they say your pistol is just a tool to get to your rifle.

I modeled this question once: Given that under stress, hands shake/move, etc., how effective is training on hitting a target in a stressful event. In other words, my brother-in-law is well trained, probably hitting the center target 9 out of 10 times at 25 yards in practice (he competes). I am not so accurate. However, under stress, most, if not all of his shots, will move off of target while it’s possible one of mine will hit target. Hmmm. Even considering wider target rings, it is possible, not so likely, that under stress I would do better than him. So, maybe pin-point accuracy in training is a little overblown.

Just a completely normal exercize in statistics. xD

Proposal for the next exercize: modeling a situation where two guys are shooting at each other and need… 3 hits?… to “win”. And if they take … 2 hits?… they “lose”. Ohh, add some more game rules and we’re getting into military science with all this! xD There’s this guy on the Internet who goes by the handle “smoothieX12” who has some videos on YouTube (and a book) about the bare basics of military science. Salvo model, Lanchester-Osipov equations, probability of a hit, and so on. 🙂

Matt Dillon scores with a quick draw and one shot from the hip. Lucas McCain sprays the scene with lead. Both methods always work for those guys. When you’re up against the Warren Oateses and Lee Van Cleefs of life, you do what you’ve got to do.

You continue to fire until you are certain the threat is neutralized.

It’s how to dispose of the bodies afterwards that is my main concern. I don’t trust “the authorities” to do the right thing in this day and age.

@Paul Murphy – The old advice is still relevant. “Keep firing until the target changes shape or catches fire.”

Joe Biden said that if you need 80 rounds, then you probably shouldn’t own a gun.

Is he right? Should there be a minimum “skill” requirement?

I’m not great at combinatorial probability stuff. My intuition tends to lead me astray, and the textbook equations typically differ from the situation at hand. For problems like this, I was taught that you first describe the situation in words. In this example, the words would be something like “in order to hit the target in N shots, you can hit the target on the first shot, or you can hit the target on the second shot, or …”. Then you examine the words you used and look for occurrences of “or” and “and”. The “or” words are a problem – for N independent events, you can’t just add up the probabilities. (If you could, then the probability of heads in two separate coin flips would be 100%.)

If there are any “or” words in the description of your problem, you have to rewrite the problem to use only “and” words. Something like “in order to miss the target in all N shots, you would have to miss the target on the first shot, and miss the target on the second shot, and …”. When you have only “and” words, you can multiply the individual probabilities to get the overall probability (for independent events).

Recently I came across an example that makes me doubt whether this rule of thumb is always true, though. It involves a game where the goal is to roll a die multiple times, add up the rolls, and hit an exact result. The player gets to pick the target number, between an upper and lower limit. So the question is what number the player should pick as the target number to maximize his chances of hitting that sum. The equations used to solve for the best number were not shown, it was just presented as a graph with a peak. But the description of how to solve the problem used “or” words, and the probabilities of individual events were simply added together. I have no reason to doubt that the graph was accurate, but I suppose that it’s possible that something got lost in the translation of how the equation was derived, to the point of an erroneous over-simplification.

Anyway, I am a bit perturbed that my precious rule of thumb may be flawed.

https://www.buckeyefirearms.org/alternate-look-handgun-stopping-power

Interesting take on cartridge effectiveness