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Old August 30, 2017, 01:43 AM   #1
JohnKSa
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Capacity, Hit Rate and Success. A Look at the Probabilities.

I've put together a number of plots that some may find useful.

How to use the plots.

The plots come in pairs. Each pair assumes at least a particular number of hits required for "success". The number of hits required for "success" ranges from 1 to 6 so there are a total of 12 plots. Although the plots are labeled with: "Success = X Hits" it would probably be more accurate for the labels to say: "Success = At Least X Hits" or maybe: "Success = X Or More Hits"

If the plot has "Hit Rate Probability" along the bottom of the graph, then each colored line on the plot represents the range of probabilities of success for a given capacity from 4 shots to 15 shots. Note that the lines on the plots are in the same vertical order as the legend on the right of the plot. You can pick a line representing a particular capacity and trace it across the graph to see how different hit rate probabilities will affect the chances of success for that capacity.

If the plot has "Number of Shots" along the bottom of the graph, then each colored line on the plot represents a range of probabilities of success for a given hit rate ranging from 10% to 90%. Note that the lines on the plots are in the same vertical order as the legend on the right of the plot. You can pick a line representing a hit rate probability and trace it across the graph to see how different capacities will affect the chances of success for that hit rate.

You can also use the plots to find the chance of success for a specific set of assumptions.

Example:

To find the probability of hitting a target at least 2 times (2 or more times) out of 8 shots with a hit rate of 50%, first find the pair of plots which are labeled "Success = 2 Hits". If you pick the plot that has "Number of Shots" across the bottom, then find where the axis is labeled 8 and trace the gridline up to where it crosses the "50% Hit Rate" line. If you pick the plot that has "Hit Rate Probability" along the bottom of the graph, then find where the bottom axis is labeled 50% and trace the gridline upward to where it crosses the "8 Shots" line. In either case, read the probability of success off the axis on the left.

The range of hits required to achieve success (1-6) is designed to cover what a person might reasonably be expected to require to solve a self-defense encounter. Might it take more or less? Of course. But going less than one doesn't make sense and making the graphs takes time so I stopped at six. If someone is really concerned about the probability for a scenario requiring more than six hits, PM me and I'll run a special case for you.

Be reasonable when you choose your hit rates if you expect to get reasonable results. There may be some of us who could really be expected to hit 80 or 90% of our shots during a gunfight, but the outcomes of real world scenarios suggest that the number is probably considerably lower. I recall reading the analysis of one study that examined a large number of police shootings which indicated that the average hit rate in a gunfight was about 3 hits for every 10 shots fired.

The graphs do not provide "high fidelity gunfight simulation numbers". They provide probabilities that are calculated based on three very simple assumptions. No more, no less. The graphs are not based on real world data or statistics, they are simply the results of probability calculations with three inputs. They can not, nor are they intended to predict the outcome of gunfights. They provide the calculated probability of "success" given an assumed hit rate probability, an assumed number of required hits for success and an assumed number of shots/capacity.


It's best to think of them as sort of "best case scenario" probabilities. The probabilities in the real world won't be better than the graphs show for a given hit rate, capacity and required number of hits, but they could easily be worse.

Here are a few ways how that could be true.

1. You get shot before you can finish firing all your rounds.
2. You fixate on one attacker and end up "wasting rounds" on him even after he's been neutralized with the required number of hits.
3. Your gun jams before you can finish firing all your rounds.
4. You never get a chance to draw and fire.

The probabilities are about finding a proper balance.

Moving up in capacity obviously improves your odds of making the required number of hits before running dry, but you can't get carried away in that direction because it's not terribly likely that a person will be able to take advantage of a huge round count in the few seconds a gunfight typically lasts.

Improving the hit rate probability (sharpening shooting skills) clearly helps a lot, but even so, if you need to make more than just a couple of hits, you still need some capacity available to take advantage of that skill. And, practically speaking, there's a limit to how much we can improve our shooting ability.

I've posted on this topic before but this is the first time I've provided such a full range of plots.

Here's one discussion of this general topic. In that thread, I only ran one pair of graphs focused on requiring at least 4 hits as the definition of success. But there's a lot of worthwhile (as well as some not so worthwhile) discussion about what the numbers do and don't mean.
https://thefiringline.com/forums/sho...d.php?t=494257

The graphs can also be useful to counter the argument that no honest person needs more than a small number of rounds for effective self-defense.

Here's another thread where some of the concepts relating to the graphs and their probabilities were discussed.
https://thefiringline.com/forums/sho...d.php?t=589112

And now, without further ado, here are the graphs.

The following two plots show how capacity and hit rate probability affect the chance of success if success is defined as scoring at least one hit (one or more hits). Note that with the higher hit rates and capacities, the probabilities compress to the top of the graph, meaning that success is very likely.


Attached Images
File Type: jpg S=1_Cap_x=HR.jpg (231.1 KB, 764 views)
File Type: jpg S=1_HR_x=Cap.jpg (228.5 KB, 717 views)
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Old August 30, 2017, 10:47 PM   #2
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The following two graphs show how capacity and hit rate probability affect the chance of success if success is defined as scoring at least two hits before running out of ammunition.


Attached Images
File Type: jpg S=2_Cap_x=HR.jpg (204.5 KB, 719 views)
File Type: jpg S=2_HR_x=Cap.jpg (193.0 KB, 694 views)
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Old August 30, 2017, 10:49 PM   #3
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The following two graphs show how capacity and hit rate probability affect the chance of success if success is defined as scoring at least three hits before running dry.


Attached Images
File Type: jpg S=3_Cap_x=HR.jpg (221.5 KB, 732 views)
File Type: jpg S=3_HR_x=Cap.jpg (208.6 KB, 715 views)
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Old August 30, 2017, 10:50 PM   #4
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The following two graphs show how capacity and hit rate probability affect the chance of success if success is defined as scoring at least four hits before running dry. This might be used to represent a scenario with two determined attackers, each requiring a minimum of 2 hits to neutralize them.


Attached Images
File Type: jpg S=4_Cap_x=HR.jpg (227.4 KB, 706 views)
File Type: jpg S=4_HR_x=Cap.jpg (213.8 KB, 711 views)
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Old August 30, 2017, 10:52 PM   #5
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The following two graphs show how capacity and hit rate probability affect the chance of success if success is defined as scoring at least five hits. Note that the probabilities start at zero for the 4 shot capacity. That's because it's impossible to score at least 5 hits with only 4 shots.


Attached Images
File Type: jpg S=5_Cap_x=HR.jpg (222.7 KB, 695 views)
File Type: jpg S=5_HR_x=Cap.jpg (213.0 KB, 667 views)
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Old August 30, 2017, 10:54 PM   #6
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The following two graphs show how capacity and hit rate probability affect the chance of success if success is defined as scoring at least six hits. Note that the probabilities are zero for the 4 & 5 shot capacities. Scoring at least 6 hits with only 4 or 5 shots just isn't going to happen.


Attached Images
File Type: jpg S=6_Cap_x=HR.jpg (213.6 KB, 672 views)
File Type: jpg S=6_HR_x=Cap.jpg (209.6 KB, 670 views)
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Old August 31, 2017, 12:51 PM   #7
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John, those are very useful indeed!

One does have to think through the meanings in terms of possible situations.

And we should consider that one may well expect to shoot three shots or maybe four at a charging attacker before pausing for a good assessment, that burns up a few rounds. We di that in a Combat Focus Shooting course, and one sees that demonstrated on such top-tier programs as
Quote:
The Best Defense.
Of course, one does not want to end up with an empty gun, and I, for one, do not want to reload publicly after the shooting stops.

A quick perusal tends to confirm the prudence of my recent decision to switch from pistol with a capacity of 7+1 to one that carries 12+1.

Thanks for taking the effort!
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Old August 31, 2017, 03:14 PM   #8
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Worth a sticky, IMO.
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Old August 31, 2017, 04:11 PM   #9
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I agree that this work deserves a sticky. I would have liked to see an area for the target size and fewer shots added but what you have done is a good coverage of average potentials for the common shooter in a self defense situation.
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Old August 31, 2017, 05:10 PM   #10
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Awesome! Sticky bait.
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http://www.teddytactical.com/archive...05_Feature.htm
Being an Academic Shooter
http://www.teddytactical.com/archive...11_Feature.htm
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Old September 1, 2017, 01:37 PM   #11
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Great work, John! Interesting to see some novel thinking and discussion too, though I know you've been working on this topic for awhile now.
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Old September 1, 2017, 04:19 PM   #12
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Great resource

Some live in states where magazine capacity is restricted to 10 rounds, 11 shots total before slidelock, anti-gunners consider this restriction "commonsense".
How might one fare with that 10 round mag limit?

Looking at the chart where success requires a realistic* 3 hits, 100% chance of success is from 10-11 rounds fired with a better than average 60% hit rate.

*Based on results from actual shootings, average 2-3 rounds required to stop an attacker:
https://www.buckeyefirearms.org/alte...stopping-power

This data is a great resource.
Should reinforce carrying more ammo than one thinks they will need.

Still, some will rationalize a snub, or 6-7 round pistol because of day/night, location, perceived risk, when they could carry something better.
Comment ^ not applicable to people who can't do better because of dress clothes at work, rather those that choose to carry "less" than they could.

Ironically, some gun owners will still rationalize less capacity than what anti-gunners consider "commonsense".
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Old September 1, 2017, 07:14 PM   #13
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Good work!
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Old September 2, 2017, 06:39 AM   #14
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Thank you!!
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Old September 2, 2017, 07:50 AM   #15
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Interesting perspective. You put a lot of work into it but I am one who having worked the streets for 30 years do not like mathematical probabilities and averages. If you are the statistical outlier..............You are a statistical outlier if you are forced to use your gun in SD. The mathematical models do not take into account any variables, only hit rate VS capacity.

I personally do EVERYTHING to improve hit rate for two main reasons, you are responsible for every bullet you fire, Where did that miss go? Bullets that miss do not help.

https://www.policeone.com/police-her...mo-on-the-job/

http://lawofficer.com/archive/office...ulis-incident/

http://www.latimes.com/local/lanow/l...htmlstory.html

I know someone will say that you cannot compare police shootings to civilian shootings. I say hogwash, one the bullets start flying they are pretty much the same.
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Old September 2, 2017, 10:44 AM   #16
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Quote:
The mathematical models do not take into account any variables, only hit rate VS capacity.
First of all, calling this a model is very misleading because it doesn't attempt to model or simulate anything. It is not a model, it is a probability calculation based on some very simple assumptions and three variables. The calculation takes the following variables into account. Hit rate. Number of hits required for "success". Capacity.

It does do not take any other variables into account because that's all it's designed to deal with.
Quote:
I personally do EVERYTHING to improve hit rate...
An excellent strategy. There's nothing in the calculation that suggests otherwise and much to indicate that strategy will increase the chances of success.

Maybe this is an easier way to think of what the graphs do.

They can't tell us if we will succeed. They can provide insight into scenarios where success is very improbable. They can give us an idea of what kinds of scenarios (based on the three parameters of hit probability, number of shots required and capacity) are "workable" and which ones aren't. They allow us to vary those three parameters to see how they affect the workability of the scenario.

If you look at the case where the number of hits required is 1, the capacity is 15 and the hit rate is 90%, the chance of "success" (making your 1 hit before the magazine runs dry) is essentially 100%. Does that mean you are guaranteed to survive a gunfight based on that scenario? NOT AT ALL! There are all kinds of way you could still be killed. Maybe your gun jams, maybe the guy hits you in the brain with his first shot before you can even fire. Maybe you slip and fall and injure yourself and end up a sitting duck. All the graph tells you is that if you fail to "win" the gunfight in that scenario, it's very unlikely that failure was the result of an unworkable combination of capacity, hit rate and the number of hits required. Something else was likely the cause of the failure.

Now let's look at the case where the number of hits required is 6, the capacity is 7 and the hit rate is 30%. We can see the chances of success are very close to zero from the graph. This tells us something very important. It tells us that we have set up an unworkable scenario. Even if we get to expend all our ammunition and everything else goes perfectly, the odds of succeeding in making all the hits we need are just about nil. If we think that this is a scenario we want to prepare for, then the graphs tell us that a 7 shot gun is a very poor choice. Well, how could we improve the situation? The graphs let us change the capacity to 15 to see if different hardware might get us out of the "doomed to fail" situation.

Unfortunately that only improves our chance of making all the hits we need before running empty to about a 3 in 10 chance of success--and that's if everything else goes perfectly. Still pretty bad. Now we should start thinking that maybe a person with the skill level to only make 30% of their shots in a gunfight is in a really, really bad situation if they need to make 6 hits. And it looks like that's true matter how much ammo they carry and even if everything else goes perfectly. We've gained some valuable perspective about the limitations imposed by our skill level and equipment choice.

Can we predict how a given real world gunfight will go? No. There are simply too many variables involved. But we can get a rough idea of what capacity level is workable for a given situation and what capacity level would put the defender seriously behind the 8 ball in that same situation or maybe even make it virtually impossible to succeed.
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Old September 2, 2017, 12:50 PM   #17
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Perhaps it is an unfortunate sign of the times. For anyone to look at look at John's analysis and expect to be able to use them to prudently choose a defensive weapon would indicate a fundamental lack of knowledge and understanding about a lot of things. And so, I think, would discounting them because they do not and cannot do that.

The analysis was not intended for that purpose.

The first and most essential step in gaining the ability to make an informed choice is training--training in shooting, tactics, and in the fundamental laws regarding the civilian use of force.

Next items on the list come from sources that may reside elsewhere.

What does one need to know?
  1. Something about how a lawful defensive use of force incident might be expected to unfold--distances, circumstances, tell-tall signs, speed..
  2. What kind of shooting skills would be appropriate, and how they differ from the suare range. Drawing fast, shooting quickly and repeatedly and effectively and sately at a target at which we had not been planning to shoot.
  3. Something about wounding effectivness.

What we learn about the first of these will never be exhaustive.

The second of these can be worked on in a good training course, or in a simulator, or probably best in FoF training. But those will not provide the reality of fear and surprise, nor will they the into account the extreme importance of avoiding hitting innocents, and most fall short in training for shooting at a fast moving targets.

But they can help us learn and prepare, and they should be useful in telling us how well we can shoot under good conditions--the maximum achievable hit rate for us.

The third starts with learning something about penetration performance. We don't have to shoot to learn that. Does our ammunition of choice meet FBI standards?

How many hits it will likely take (or may take) will be a guess, but it can be an informed guess. Massad Ayoob recommends that people keep on hand a copy of Gray's Anatomy and refer to it from time time. I would add that studying the FBI report on handgun wounding effectiveness--the medical parts--can help us to better understand that anatomy book, at least to some extent.

The above should give one some basis for understanding the envelope, as it were, for reasonable assessments of hit rate and of the number of hits required.

Then, and only then, will you be able to apply John's graphs in the making of a selection.

Actually, you should have been able to take some guns off your list early in the process.

That's not all, however. Obviously, the gun that might make you most effective might, and probably will, be too bulky and heavy for reasonable everyday carr,

Back to the judgment call.
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Old September 2, 2017, 05:11 PM   #18
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Interesting links Nanuk! What amazes me is most of these shootouts were huge number of rounds are fired, the participants don't realize their methods are failing and shift target priority (as in head shots.) The Mozambique has fallen out of favor I guess.

As for mathematical models, since I don't really care what the populace carries or does, it does not matter much to me.

It's what I can do, what I carry, how I've trained myself that matters.

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Old September 4, 2017, 01:11 PM   #19
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I'm looking forward to reading that. I can tell that you aren't doing anything predictive. It's interesting looking at statistical analysis, too.

Forty years ago I started reading about chaos theory, and moved on to realize that every tiny detail can make or break an outcome, that probability is almost useless in planning ahead.

Chaos truly is the king. Put a guy shooting hoops at seventy feet and calculate the chances of random hits. After five hours you would expect fatigue to screw up everything, but it would still be a beautiful curve downward until he started wearing out so badly that he couldn't reach the basket and it drops to zero.

But all it takes is a few drops of sweat to screw up a shot.
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Old September 4, 2017, 02:09 PM   #20
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Quote:
I know someone will say that you cannot compare police shootings to civilian shootings. I say hogwash, one the bullets start flying they are pretty much the same.
The differences lie in what leads up to tho shooting, and in what might happen after what a civilian may experience.

Beforehand:
  • A sworn officer is expected to intervene, when necessary, to maintain the peace.
  • A sworn officer makes traffic stiops, and the subjects may well be violent wanted criminals.
  • Sworn officers go into bars to stop fights.
  • Sworn officers respond to calls regarding domestic violence.
  • Sworn officers may enter structures to check for suspects.

Civilians should be doing none of those things.

The differences are relevant to the distances and to some other aspects of the encounters.

Afterward:
  • Once the civilian has sucessrully devended himself, it's over; the sworn officer is expected to pursue the criminal.
  • The sworn officer takes supects into custody; the civilian is not expected to do that.
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Old September 4, 2017, 05:04 PM   #21
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You should keep in mind two things. Police officers are dependent on and skilled at situational awareness.

Police officers are trained to some degree, I believe that almost every police organization will require every sworn officer to qualify. A police officer that can't serve in an emergency situation, even if his only responsibility is the evidence locker is a waste of resources.

With the literally hundreds of thousands of people who walk around half drunk, texting, absorbed on their company or carrying a burden, none of these hundreds of thousands of people could properly respond, especially if their training is inadequate. To aggravate the situation, these shmoes who wander in a daze are targeted. They pop out their seven shot .380 and are at a huge disadvantage. I hope that no cop on the planet is as lame as many of the concealed carrier. heck, I hope that they are better than me. I have weak spots.
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Old September 8, 2017, 11:13 AM   #22
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I care far more about how many times I might get hit than how many times I might hit a bad guy. My getting shot would screw up a lot of good fishing.
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Old September 8, 2017, 09:20 PM   #23
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In my opinion you are way over thinking this. I don't think you can calculate the outcome of a gunfight in any meaningful manner. There are WAY too many variables.
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Old September 8, 2017, 09:33 PM   #24
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Overkill777,

The only metric that matters is surviving.
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Old September 9, 2017, 07:55 AM   #25
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Quote:
In my opinion you are way over thinking this.
In what way?

Quote:
I don't think you can calculate the outcome of a gunfight in any meaningful manner.
Has anyone suggested otherwise?

Quote:
There are WAY too many variables.
Yep.

John put it this way:
Quote:
Maybe this is an easier way to think of what the graphs do.

They can't tell us if we will succeed. They can provide insight into scenarios where success is very improbable. They can give us an idea of what kinds of scenarios (based on the three parameters of hit probability, number of shots required and capacity) are "workable" and which ones aren't. They allow us to vary those three parameters to see how they affect the workability of the scenario.
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