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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