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Old June 27, 2012, 11:48 PM   #1
JohnKSa
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Capacity, Hit Rate, Multiple Assailants and some thoughts...

A thread on another forum got me thinking about the probability of making a certain number of hits within a certain number of shots given a fixed probability of hitting the target with any given shot.

I set up a spreadsheet to do the calculations and thought some of the results might be interesting to other shooters.

To present the data in some sort of reasonable fashion, I've assumed that an assailant will require 2 or more HITS from a handgun to be neutralized/incapacitated. It's fairly common to assume that it will take more than one hit to neutralize an assailant and since double-tapping is ubiquitous, I figured 2 was a reasonable starting assumption.

I've assumed a hit rate probability of 30% for the listings below since that is an often quoted figure for the hit rate probability of law enforcement officers involved in gunfights.

For a single assailant and a 30% hit rate probability.
# of Shots : Probability of achieving 2 or more hits.
5 : 47.2%
6 : 58%
7 : 67.1%
8 : 74.5%
9 : 80.4%
10: 85.1%
11: 88.7%
12: 91.5%

For two assailants and a 30% hit rate probability.
# of Shots : Probability of 4 or more hits (i.e. 2 on each assailant).
5 : 3.1%
6 : 7.1%
7 : 12.6%
8 : 19.4%
9 : 27%
10: 35%
11: 43%
12: 50.8%

There are some other assumptions inherent in trying to apply these probabilities practically. For one thing, the two assailant case assumes that the defender is able to tell how many hits have been made on the first assailant and then immediately switch to shooting at the second assailant after making 2 hits on the first--wasting no additional shots on an already neutralized opponent.

Both cases assume that the defender is able to empty his/her weapon in the course of the gunfight--he/she is not incapacitated before that can take place.

Basically, these are sort of "best case" scenarios. The point is to get a rough idea of the best that a gunfight could turn out if you need 2 hits per assailant and you have a given number of shots to pull it off.

I was surprised at how tough it was to neutralize 2 assailants given a 30% hit rate and 5 shots. Basically one can expect to succeed 3 times in 100 attempts.

To improve those odds to EVEN odds (roughly a 50/50 chance of success) when using a 5 shot handgun, one would need to shoot well enough to achieve a 69% hit rate during a gunfight. A 69% hit rate would give a person with a 5 shot handgun a 50.1% chance of succeeding against two opponents who each require 2 hits to be neutralized.

From a practical standpoint, the probabilities involved suggest that someone armed with a typical small carry pistol (11 rounds or less) and achieving a hit rate of about 30% per shot has better than even odds of failing to neutralize 2 opponents before their gun is emptied. Under the same conditions, someone armed with a true pocket pistol (7 rounds or less) is likely to fail to neutralize 2 assailants about 90% of the time or more.

Even with only a single assailant, a pocket pistol will run dry somewhere between 1/3 and 1/2 of the time before a 30% hit rate achieves 2 hits.

Lance Thomas realized after winning his first gunfight with a 5 shot pistol that he had expended 3 of his 5 rounds neutralizing one opponent--fortunately the other one ran. The realization caused him to change his tactics to include multiple guns in his defense plan, hidden behind the counter in various locations around his gun shop. That's one fairly practical response.

However, I'm not really suggesting we all need to carry multiple guns or upgrade to high-capacity carry guns. The major "takeaway" from the calculations is to understand the limitations of the weapon system that is the combination of you and your carry gun.

Taking on 2 determined attackers with a typical compact pistol is a pretty grim mission, if you look at the numbers. If one or both assailants don't cut and run when the lead starts flying, the odds are slim to none of success. If success (survival) is the goal, it might be wise to consider other options. Drawing and shooting it out isn't going to be a wise course of action unless you're sure they'll give up easily or unless there is no other reasonable course of action available.
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Old June 28, 2012, 12:00 AM   #2
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While your results are not really surprising they are a little sobering if you really think about it. The first thought is that I need to practice more, but really I am a pretty good shot. The issue is not skill to shoot in a controlled environment, but the ability to respond in a high stress situation. The second thought is maybe I better find a way to carry the P229 instead of the J-Frame.
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Old June 28, 2012, 06:44 AM   #3
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Capacity

No expert here - just someone that shoots a lot and has for a long time. I really like capacity more and more. I find it difficult to carry a 1911 (have 3 custom/semi custom) because of the capacity. I also have been practicing high speed shooting at close range ie 8 yards. When you can fire 6-11 rounds in one second and all are in the chest zone, high capacity makes sense to me. I'm not so sure that capacity may not trump everything in self defense carry. I also sometimes think that pin point accuracy is the enemy of defensive shooting. If I shoot slowly I can shoot about 1-2 inch groups at 8 yards. If I shoot 6-7 rounds in one second the group will be 6-7 inches. I suspect that the 6" group is better in a self defense scenario.
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Old June 28, 2012, 07:19 AM   #4
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Capacity

Just an aside to this topic. I bird hunt with a Gentleman that was awarded a Silver Star in Vietnam. He seems about as far from a hero as it's possible to get. I told him once that "you just don't look like a hero to me".

He replied "I'm not. I was a 19 year old kid, that the worst helicopter pilot in the world, landed on the top of the wrong mountain with a bunch of other kids. Then couldn't come back because of weather and pick us up. I had a position on the line that the enemy tried to come through at about 3 am. All I did was fire five 30 round magazines in just a very very few minuets. I was just trying to keep them off of me. If the rifle had jammed, or ammo ran out - I wouldn't be here."

That influences the way I think about "capacity" anymore. It's just a way to "keep them off of me". I also asked myself that question when buying "keeper" AR rifles. I asked "Will this rifle fire five 30 round magazines in 5 minuets and still work?". A military Colt will do that, along with FN and perhaps others. At any rate I ended up with a FN and Daniels Defense AR's. After using them I believe both would run until they melted, if I ever needed to do that. Just an old mans ramblings. Your mileage may vary.
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Old June 28, 2012, 07:45 PM   #5
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Personally, I think this thread should be stickied. It would be useful for any number of ongoing debates.
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Old June 28, 2012, 08:26 PM   #6
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I have absolutely no experience in shootouts, and I hope to maintain that record for the rest of my natural life. Note the emphasis.

I respect the work done by the OP, and with a fair bit of experience in higher math, statistics and analysis, I have no quarrel with his methodology.

All that said...

I have seen zero evidence that the actual, documented experiences of CHL holders (or SD experiences in general) bears out this thesis.

In fact, quite the opposite.

I am open to evidence to the contrary, and welcome it, as my choice of concealed carry weapons has gone through quite a journey, and is still evolving...but has remained somewhat stable over the past three years. These days, I carry what I shoot best (either a 6-shot S&W revolver or a 1911).

If my current choice is the wrong one, and this can be proven...I am open to change. But it's going to require some fairly solid evidence to change my mind...

Best regards, Rich
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Old June 28, 2012, 08:58 PM   #7
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@orionengnr: I am not sure I completely follow your response, but, what I think I hear you saying is, there is more to think about than capacity. If so, I'd agree with that, and don't think that is necessarily at odds.

The math of the OP is telling me - if all else is equal, more capacity gives you a greater likelihood of making the stop. I think that's pretty hard to argue with.

You though, seem to be saying that, all else may not be equal, i.e. you might have a hit probability of 0.25 with gun A and 0.50 with gun B that you are very familiar and confident with, that has excellent sights/practical accuracy. Then even if gun B has fewer shots, it might still be your best bet. That can also be true.

So here's two other calculations (1 assailant):
1. Hit probability per shot = 0.25, 10 shot capacity, probability of two or more hits when empty, is 0.76

2. Hit probability per shot = 0.50, 5 shot capacity, probability of two or more hits when empty, is 0.81

So yes, *if* you shoot much better with one gun than another, that can outweigh a capacity disadvantage.

See free online binomial distribution calculator at:
http://stattrek.com/online-calculator/binomial.aspx
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Old June 28, 2012, 09:15 PM   #8
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I applaud the work of the OP.

I am not a mathematician, but a cop.

I have seen people dead of one shot, I have seen an officer that was shot up so bad he was bleeding IV fluid and lived. I have seen people fire one shot and its over, there was the famous FBI Miami shootout, where one Agent fired 15 shots laying across the hood and missed them all from 3 feet. Statistics are worse than anecdote.

There are too many variables, how many BG's have soaked up 8-10 shots from 9mm, 45, 357 mag and still kept fighting? It is a sobering read and interesting in that, the case you are presenting is capacity trumps all. There are a lot of assumptions in your argument. Stress is just one thing to add to the pot, you may be diving for cover, shielding your daughter or what have you.

How do you train? How do you practice? Proper training and practice will sway the odds in your favor. I am a big advocate for point shooting - not hip shooting. Bring the weapon to eye level, look where you are shooting, muscle memory and peripheral vision will bring the shots in. Point your finger, look where you are pointing, it is that easy out to 15 yards.

http://www.jimgregg.net/
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Old June 28, 2012, 09:51 PM   #9
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I agree, the more rounds I have in the gun the better, of course my daily carry is almost always a pocket gun with 7 or less rounds in it. I can't carry anything bigger at work.

Here are a couple of great videos from a great instructor.

http://www.youtube.com/watch?v=2dA36NYLqns
http://www.youtube.com/watch?v=DXohG...feature=relmfu
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Old June 28, 2012, 11:46 PM   #10
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Quote:
I have seen zero evidence that the actual, documented experiences of CHL holders (or SD experiences in general) bears out this thesis.

In fact, quite the opposite.
You noticed that too.

The only conclusion I can draw is one we should already have expected. Defenders with low-capacity handguns who prevail against multiple attackers are NOT doing so by shooting all their attackers to the ground by making multiple solid hits on each one.

Clearly, what's happening is that one or more of the attackers is chosing to run rather than stand and fight. The Lance Thomas incident is a perfect example. Mr. Thomas, armed with a 5 shot handgun, prevailed against two attackers--but not because he applied 2 solid hits each. He won because the second attacker ran when the shooting started.

So, does that mean that capacity is meaningless? Not really. It was simply the luck of the draw that the second man ran. Had he stood and fought like his accomplice, Mr. Thomas would likely have not prevailed given that it took him 3 shots to neutralize his first opponent, leaving him only 2 to deal with the second.

Clearly there needs to be a balance.

Moving up in capacity obviously improves your odds, but you can't get carried away in that direction because it's not terribly likely that you'll 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, but only if you have the capacity available to take advantage of it. For example, even a very impressive 70% hit rate only gives you a 53% chance of scoring 2 or more hits on each of 2 opponents if you're armed with a 5 shot handgun. On the other hand, if you can achieve just a 50% hit rate with a 9 shot handgun, your odds of success are 75%.

As with many things, it's a tradeoff--a balance needs to be found.

Clearly these calculations don't tell the whole story. They only provide limited insight into certain aspects of a gunfight. That insight needs to be combined with other information before the big picture can start to take shape. But without that insight, a person can have a very mistaken impression about their chances against more than one determined attacker.

Here are some plots that lay things out fairly well.

In this plot, each line traces out the probability of success with a given hit rate. So if you take the bottom line (10% hit rate) and trace across to where the bottom axis label reads 9 (# of shots), the height of the line will give you the probability of making 4 hits with 9 shots if your hit rate is 10%. It's a very small number...


In the plot below, each line traces out the probability of success with a given number of shots. So, if you take the bottom line (5 shot line) and trace across to where the bottom axis label (hit rate) reads 50%, the height of the line (about 20%) tells the probability of making 4 hits with 5 shots and a hit rate of 50%.
Attached Images
File Type: jpg Success vs # of Shots_Small.jpg (125.9 KB, 895 views)
File Type: jpg Success vs Hit Rate Probability_Small.jpg (125.5 KB, 882 views)
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Old June 29, 2012, 09:40 AM   #11
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I still think this is flawed data. You cannot say with any degree of certainty that 4 hits are going to neutralize every enemy, so how can you give it a 100% probability of incapacitating an opponent? Sounds nice for a video game, but real life does not work that way. If that was the case I would carry the Beretta 84 (14 shot 380). This data does not take into account the effectiveness of the projectile being used, it lacks any credibility.

For what its worth, back in the revolver days, I was taught on multiple assailants to put one in each one, starting with the highest threat and return to those that need followup.

Tactics usually play a significant role in the outcome of a gunfight, cover, cover,cover.
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Old June 29, 2012, 10:28 AM   #12
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Quote:
Posted by Nanuk: I still think this is flawed data. You cannot say with any degree of certainty that 4 hits are going to neutralize every enemy, so how can you give it a 100% probability of incapacitating an opponent?
One cannot reasonably characterize these calculations as "flawed data" on that basis.

The analysis defined success as two hits. It's an arbitrary but fairly reasonable assumption, chosen for purposes of illustration.

And there are a number of other assumptions, all clearly stated.

What the analysis does is illustrate, for those who may not have really thought about it, the likelihood of being able to achieve success with a certain number of rounds, (1) with an assumed hit rate, (2) with the shooter being able to stop shooing after a second hit, (3) with the shooter able to continue shooting rather than being knocked down or incapacitated, and, in the case of a second attacker, (4) assuming that the second attacker continues the attack.

It is a real eye-openter.

Flaws? You can vary the assumptions. The one that would seem most tenuous to me is that the shooter will be able to stop shooting after achieving two hits. Most people are trained to fire very fast, and would likely have fired several times before ever detecting a second hit, should a second hit actually do the trick.

There's a lot of debate about the fourth assumption. All of my friends who carry five or six shot firearms say with a lot of expressed confidence that a second attacker will most certainly turn tail and run after the shooting starts. That may happen, or it may not. There's a lot of wishful thinking in that, and more than a little justification of having chosen to carry a pocket firearm, not to mention the fulfillment of a natural need to feel safe while doing so.

If one is talking about a robbery where the taking of money or jewelry is the objective, and if the second robber can, if he exits the door, go away to rob another day, he will probably try to do so. If the crime is a carjacking, and the perps desperately need a different, operational, and fueled automobile in order to make their escape, I would not make that assumption. Somewhere in between is the perp who thinks he has a greater chance of surviving by continuing to close a short distance than by trying to run.

Quote:
This data does not take into account the effectiveness of the projectile being used, it lacks any credibility.
You have completely misinterpreted the point of the analysis. The "credibility" resides in the math; you plug in your own assumptions. If you want to vary the number of hits that would constitute success, you can do the math yourself. One might be well served to define three hits as success.

There are a lot of people who reflect upon their groups at the range, assume that their firing rate is realistic, and perhaps also making overoptimistic assumptions about wounding effectiveness, conclude that they are well armed for whatever might befall them.

They can learn something by doing two things: (1) getting off their first and subsequent shots much more quickly--say, 1.2 seconds from signal to first shot and continuing at four or five shots per second to get an idea of what their real hit probability may reasonably be; and (2) putting a little thought into John's analysis.

Actually there is a third thing: having a willingness to discard preconceived notions that don't look could in the light of the results.
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Old June 29, 2012, 10:57 AM   #13
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I have a question, and I apologize if it comes off as persnickety..

Is there any data available to compare recent hit percentages with those from when revolvers were the primary issue weapons?

The reason I ask is that it seems possible that part of the reason percentages are so low is that police officers - who I assume are the source of most of the data - have access to so much ammunition.
Every time the NYPD shoots 80+ rounds in a suspects general direction, or uses "covering fire" it's got to sway the data. I don't know how significantly it would alter it as I don't know how often that actually happens, but it seems like it could be a factor.

I would assume that someone with a 5 shot revolver would be less likely to shoot at a rate that would lead to 30% accuracy. But I could certainly be wrong.
That's not to say I would rather have 5 shots than 15, or that people necessarily shoot autos too fast, but it does seem like a statistically significant amount of the data used to get the 30% hit rate might not be aimed fire.
BUt maybe that doesn't really matter.
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Old June 29, 2012, 11:37 AM   #14
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Very good analysis JohnKSa.

Quote:
I still think this is flawed data. You cannot say with any degree of certainty that 4 hits are going to neutralize every enemy, so how can you give it a 100% probability of incapacitating an opponent? Sounds nice for a video game, but real life does not work that way. If that was the case I would carry the Beretta 84 (14 shot 380). This data does not take into account the effectiveness of the projectile being used, it lacks any credibility.
I think you are looking at it wrong.

No you can't say with any degree of certainly that an number of rounds will stop someone unless it hits some very specific targets. But you have to "draw a line in the sand" somewhere and build an analysis from that. Things can be tweaked as needed to make the simulation more closely reflect reality.

I don't think John is saying that this is "reality". What he is saying, is that given these reasonable assumptions, here is what you statistically get.

I think this is good information to take to the range to test. People should take these assumptions and try them out. Put 2 anatomical targets out and do some draw from concealment shooting as quickly as they would in a defensive situation at reasonable self defense ranges and see how they do compared to this data.

If they have one or more rounds in a vital area on their targets and they believe that they can do that on command on moving targets, they are good to go. If, however, they can't replicate Johns data on even a square range on non moving targets, they might need to re-evaluate their tactics/training or equipment. (In that order).
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Old June 29, 2012, 11:41 AM   #15
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Quote:
Posted by dayman: ...it seems possible that part of the reason percentages are so low is that police officers - who I assume are the source of most of the data - have access to so much ammunition.
Interesting question. Two things:
  • Police oficers and civilaians with training are trained to try to put sufficient rounds into a dangerous attacker as quickly as possible; this isn't because they carry so many rounds, but because handgun bullets are not all that effective.
  • We really shouldn't be making any assumtions from police stats (not very applicable) or from civilian stats (far too few data available); we need to assess likely hit percentages from realistic training with a lot of rounds fired.

It is not that police officers fire a lot of rounds because they carry the ammunition; they carry the ammunition because of their need. When some departments switched to .40 cal pistols from 9mm, which had higher capacity, they added magazines to their belts to make up for the difference.

Quote:
I would assume that someone with a 5 shot revolver would be less likely to shoot at a rate that would lead to 30% accuracy.
I would assume that regardless of what one carries, one will fire whatever it takes as quickly as possible.
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Old June 29, 2012, 01:48 PM   #16
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So given "average" hit probability, a 10 shot subcompact (Glock) is a good starting point (capacity wise) for CC and surviving a two person attack?

How about if the defender is also carrying extra 9 - 10 shot magazine?
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Old June 29, 2012, 01:54 PM   #17
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I think that was an excellent question, although it may be of more historical interest now than of practical value, referring here to revolvers and hit ratios. I've also seen the question brought up before.

What I've read suggests that revolver-armed policemen fired more slowly, though I don't recall if their results were any better or not. But what is interesting, if you go back far enough, is the variety of shooting methods officers have been taught to use. While one-handed shooting was usually assumed, two-handed shooting stances were expected to be used "when necessary" even before WWII. The one-handed style probably still dominated the customary combat positions for a long time. Even that was often what we call "point shooting" sometimes, even in big city departments where there actually was training.

It is also interesting to note the influence of shooting games on real-world self-defense training, for better of worse, although I'd have to say it has more recently been more positive than not. In the late 1940s and 1950s, Western style fast draw was even seen to have some value for real world application. That very fact should make you think twice about the value of contemporary shooting games.

Finally, Fairbairn, in his book "Shooting to Live" stated that the more his gun resembled a machine gun, the more he liked it.
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Old June 29, 2012, 02:25 PM   #18
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Quote:
Posted by CDW4ME: So given "average" hit probability, a 10 shot subcompact (Glock) is a good starting point (capacity wise) for CC and surviving a two person attack?
Are the other assumptions valid? Do you really think that a 30% hit probability applies? I'm not trying to be difficult, but we cannot get too elegant or 'scentific' in solving a very subjective problem with a lot of unknown variables. What John's calculations do for us is provide some very illustrative, thought provoking, and perhaps surprising food for thought. And it should destroy some misconceptions. But it's not intended as an accurate prediction model.

I do not know why a ten shot pistol would not be a good idea.

Quote:
How about if the defender is also carrying extra 9 - 10 shot magazine?
The police carry several magazines, but remember, they chase people; they press the attack; they do not try to retreat.

For the civilian, the extra magazine serves as much to solve a malfunction as anything else.
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Old June 29, 2012, 05:33 PM   #19
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Charts and speculation don't mean jack in the Real World. Friend of mine was involved in a real gun fight in Venezuala (which means he's been in one more gunfight than some so-called s.d. "expert" seminar givers), in which the shooting started at about 40 yds. Friend was hit first by assailant's .357 mag revolver and friend went down. Fortunately for friend, and considering that there were four assailants, friend was carrying a G17, not something of lower capacity.

Friend is still alive. At least one of the assailants isn't.
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Old June 29, 2012, 07:27 PM   #20
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OldMarksman, the only problem with basing percentages on rounds fired in training, is that most of us don't train with anybody shooting back at us.

I've read one article that indicated a fair percentage may put their first round or two into the ground, as their weapon comes up from the holster, due to adrenaline reactions when they are taking fire. (The article was on police involved shootings, and the people putting rounds in the ground were officers.)

I'm not sure what kind of training would realistically create both the adrenaline-based physical effects and the mental fear effects of taking fire.

Were I to base my hit percentage on normal hit rates on human sized targets from 3-15 yards, I might assume virtually 100% as a percentage on static targets, and somewhere around 80% on targets that are moving at a walking pace. (Actually, the moving targets I've shot were either on a skateboard, or mounted on swingers.)

I seriously question whether I could come close to that if bullets were coming back my way. So, I figure the 30% police averages are probably realistic. I suspect I'd be very happy to achieve 50% if I were taking fire.

YMMV.
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Old June 29, 2012, 08:33 PM   #21
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Quote:
Is there any data available to compare recent hit percentages with those from when revolvers were the primary issue weapons?
That's a good question, and I don't have an answer for it.

However, we can look at the "revolver-like" scenarios to see how a higher hit rate affects the outcomes.

If we look at the second chart and focus on the 5 and 6 shot traces, we see that it takes a hit rate substantially exceeding 50% before the chances of success rise to a 50/50 chance of success. If we want to attain a 70% success rate it takes a hit rate better than 65% with either 5 or 6 shots.
Quote:
Charts and speculation don't mean jack in the Real World.
They can mean a great deal.

In this case, what the charts mean, in the real world is this:

If you have a 5 shot handgun and if your effective hit rate is 30% in a gunfight and if you are faced with 2 attackers that each require at least 2 hits each to be neutralized(stop attacking).

Then your chances of success (success=making 2 or more hits on each of your attackers) do not exceed 3.08%; your odds of failing are 97% or worse.

Your chances of success might actually be worse than 3% in the real world for any number of reasons that aren't foreseen by the relatively simple assumptions made to govern these calculations, but they can't be any better, within the bounds of the assumptions lined out in the previous paragraph.
Quote:
But it's not intended as an accurate prediction model.
Correct. The charts give only a kind of "best case" outcome based on the stated assumptions.
Quote:
One might be well served to define three hits as success.
The plots pretty much tell the whole story for the assumption that 4 hits are required, but I can run different scenarios now that I have the spreadsheet set up.

Making a new set of plots takes a bit of time, so I'm not offering to do that, but if anyone has some numbers that they want run based on a different number of hits required for "success", I'll be happy to run the numbers for some different scenarios.
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Old June 29, 2012, 08:57 PM   #22
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Quote:
Is there any data available to compare recent hit percentages with those from when revolvers were the primary issue weapons?
The last year the USBP carried revolvers the hit ratio was 80% the first year we carried semi auto's it was 20%. I have no idea what the latest scores are.

If you want to know how YOU are going to do, try an IDPA match. At least you will be doing more than square range tactics and bragging about your group.

I was very comfortable with a revolver in law enforcement for many years, as long as it was a magnum. It used to be that 1 357 hit equaled 3 from a 9mm or a 38 for effectiveness on the target.

It is interesting in the academic, but that is as far as it goes. Basing this theory of effectiveness on a mathematical formula based on statistics and assumptions is all fantasy warrior-ship.
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Old June 29, 2012, 09:01 PM   #23
jason_iowa
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As a civilian or as civilian le I have never had to fire my gun and only in the last year have I had to draw it. I carry a single stack 45 most of the time + a 357 snub 1 extra mag for the 45. I really don't need the reload if I had an issue with the auto I would ditch it and go to the revolver. I feel completely comfortable with 5 or 6 rounds with up to 3 people. I have had hundreds of hours training combat shooting + idk how many hours of actual combat. At hand gun ranges I'm not worried about missing at all let alone 2/3rds of the time. Practice makes perfect.
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Old June 29, 2012, 09:09 PM   #24
Shotgun693
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Avoid high crime locations, don't flash a lot of money, don't frequent prostitutes, don't do dope, don't go to seedy bars, don't brag about the vast gun collection at your home. Doing these things will do as much to keep you alive as carrying a gun of any type.
Oh yeah, if there's a Gang Member in your family throw them out of your house.
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Old June 29, 2012, 09:10 PM   #25
JohnKSa
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Quote:
Basing this theory of effectiveness on a mathematical formula based on statistics and assumptions is all fantasy warrior-ship.
The formula/calculations have no bearing on effectiveness in the sense of predicting a person's success in an actual gunfight.

There's really no way to take the results and turn them into some way of predicting how well someone will do in a gunfight the real world. That's because success for the purpose of the calculations is defined as making a certain number of hits and we all know that it takes more than simply making a certain predefined number of hits to insure success in a real-world gunfight.

What the charts effectively do is tell the probability that a defender will make 4 or more hits with a given hit rate probability and a given number of rounds available. They will provide that information reliably and accurately.
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