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Hey everyone,
So as we all know, mathhammer gives us a way to compare the effectiveness of different items in the game. Many times, this is used as a simple "odds of killing X with Y". However, I wanted to expand upon that slightly.

Here, I wanted to address the issue of "potential". For example, while a railgun can get a penetrating hit on AV14, an assault cannon can theoretically get 4. So, the following is a sample of a couple different weapons against a space marine and an ork. The idea is to give players an idea of the ability of weapons to inflict multiple wounds on a given unit to further compare their effectiveness. I have graphed these as a probability distribution. Feel free to discuss.

The basic formula used is (shots) x (to hit)^n x (to wound)^n x (to not save)^n where "n" is the number of desired wounds.

Some other points:
200% to inflict 1 wound does not necessarily equal 100% chance to inflict two wounds due to the stacking of probabilities
In a given game, you will generally not roll enough dice to balance out to statistical probabilities. Thus, the average game will involve more luck than probability.

Disclaimer: These are not probabilities (despite the label of the y axis, I couldn't think of a better term). You cannot have a probability greater than one. Rather, the numbers produced are ratios of effectiveness of a weapon against different units. True, standard deviation and rigorous application of probability rules will get you a precise answer, but a comparison of ratios is sometimes sufficient for normal play. Mathhammer itself is an application of expected results, this is merely an expansion of basic mathhammer.

Perhaps I am wrong, but 'potential' is irrelevant for pure mathhammer. Merely the statistically average sucess. Sure, on one end of the spectrum extreme gain is possible, but it is evened out overall by the other end. What the system of potential is doing is merely charting additional possibilities instead of simplifying it further. These possibilities can be averaged out to come up with the traditional solution.

Actually in 40K potential could be argued as being more important than stat avg success.

Think of it this way if I have the chance to blow up the opponents key unit in the first turn rather than an average of blowing it up in four turns is the gamble worth it? The answer is perhaps depending on the odds but you will never know those odds if you don't look at potential.

In a game where the order of events doesn't matter potential doesn't matter but that isn't 40K.

That is why assault cannons and lascannons are so fantastic cause they can literally kill anything with the right roll.


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Whoops, hit enter before asking my question.

Is there any special meaning behind the weapons chosen for this example?

Or did you have a particular formula or calculation method or example to show?

I agree potential is often overlooked but I think that that is often due to the added steps needed to determine a potential distribution.

In probability you can never have over 100%

For example. Ork Vs Boltgun (BS4)

1 event 2 independant dice (no armor save)

36 outcomes-
2/3 of first roll 1/2 of second roll is number of unsaved wounds.

1 shot 33% chance to kill an ork

2 shots is not a 66% chance to kill an ork.

4 dice 1296 combinations.
die one and two will be the same. You have 4 possible outcomes with 2 sets of dice.

1: no wounds
2: 1 wound from the first set
3: 1 wound from the 2nd set
4: 2 wounds.

I'll build the set later, but I hope that presents the idea.

Potential is nice to know and useful for analysis, but is much harder than mathhammer-probability to drum up mid-game and use practically.

welcome to the world of standard deviation.

Fair point, but would be nice if you took your expanding just a tad further.

A quick reference probability table would be awesome

Needs some thinking how you would lay it out, for me it's by armour grade and toughness, so it's actually a piece of work required but we could all use it.

I contend the easiest way is standard deviation. Probability distributions are great, but then it feels like I'm doing financial analysis instead of playing 40k. Plus, people like to use the average method, and standard deviation works well with it.

You know within +/- one standard deviation, you have a 67% chance of something within a bell curve happening. Within 2 standard deviations is 92%.

So you might have something like...when 10 marines charge on average they do 5 wounds +/- 2,

That allows you to do a quick worst case/best case/average case analysis without really slowing down the game.

You can also automate a lot of the calculations ahead of time, since d6 probabilities are finite and known.

Wait, you're telling me that you guys actually stop ingame and go over the statisticall chances? I do this too... during list building...


Anyway, sorry for derailing the topic. I agree; potentional dammage is an often overlooked part of mathhammer. The sole reason that Orks are so incredibly nasty is that while they may cause as much average dammage as a marine army, their potentional dammage is much higher.

You can do most of it in your head...you don't need to be precise. Knowing 3.56787 whatever decimal point isn't helpful, but knowin'g it's a number between 3 and 4 isn't hard to figure out. If you've looked at the standard deviation for common situations, you'll have a rough idea worse case/best case and you can make you decisions rather quickly.

 TheCaptain wrote:
Potential is nice to know and useful for analysis, but is much harder than mathhammer-probability to drum up mid-game and use practically.

This. There's all kinds of math you can do to analyze something (probability of getting at least X, expected range with 50% probability, etc), but just calculating average outcomes is usually a close enough approximation and a lot easier to do.

There is absolutely NO way that you can get a probability above 1. You have made a mistake in at least three of your calculations, and I imagine in the fourth one as well.


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 Peregrine wrote:

 TheCaptain wrote:
Potential is nice to know and useful for analysis, but is much harder than mathhammer-probability to drum up mid-game and use practically.

This. There's all kinds of math you can do to analyze something (probability of getting at least X, expected range with 50% probability, etc), but just calculating average outcomes is usually a close enough approximation and a lot easier to do.

Actually, just calculating average outcomes will give you a terrible idea of what will actually happen in a small sample setting (like a single game). How many times are you going to get 0.2 or 0.7 wounds on a squad in a round of shooting? Never.

The calculation might be easier and the model simpler, but that does not make it a good approximation.

Standard deviation is a limited tool is this application. The standard rules of thumb, like 65% within 1 standard deviation, rely on a normal distribution. Dice are basically always going to be a binomial. While the binomial converges to a normal as the sample size gets big, you won't get there in most applications.

And the charts in the OP are clearly wrong, as others have noted.

The bottom line is that to truly cover the full spectrum of possibilities, you need the full probability distribution, which is inherently hard to summarize. Anything else, like the mean plus SD combo, will leave stuff out. So there's no easy way to sum up mathhammer results.

 Barksdale wrote:
Actually, just calculating average outcomes will give you a terrible idea of what will actually happen in a small sample setting (like a single game). How many times are you going to get 0.2 or 0.7 wounds on a squad in a round of shooting? Never.

But that's not the intent. Calculating the average outcome isn't about getting a precise answer, it's about getting a rough estimate. If you're doing math in 40k it's almost always for one of two reasons:

1) You're in the middle of a game and need to plan your strategy. You don't have time to get out the calculator and figure out the exact probability of getting {0-10} dead marines with an IG veteran squad armed with plasma/lasguns/autocannon, you need a quick answer. So you calculate the average outcome for your unit. If you get an answer of 0.2 you can pretty safely assume you're wasting the unit's fire. If you get 0.7, you know you might do a bit of damage but shouldn't be surprised if you fail entirely. If you get 2.7 you can assume you'll probably get about 2-4 wounds. Etc. Any semi-intelligent player knows to plan for the possibility of exceptionally good or exceptionally bad rolling, so the extra information from a more detailed analysis doesn't really add much to your planning ability and takes a huge amount of extra time.

or

2) You're not in a game and are comparing two units. In this case comparing the average results for both units usually gives you enough information about their relative effectiveness, a more detailed analysis usually isn't going to add very much to the discussion.

 Peregrine wrote:

1) You're in the middle of a game and need to plan your strategy. You don't have time to get out the calculator and figure out the exact probability of getting {0-10} dead marines with an IG veteran squad armed with plasma/lasguns/autocannon, you need a quick answer. So you calculate the average outcome for your unit. If you get an answer of 0.2 you can pretty safely assume you're wasting the unit's fire. If you get 0.7, you know you might do a bit of damage but shouldn't be surprised if you fail entirely. If you get 2.7 you can assume you'll probably get about 2-4 wounds. Etc. Any semi-intelligent player knows to plan for the possibility of exceptionally good or exceptionally bad rolling, so the extra information from a more detailed analysis doesn't really add much to your planning ability and takes a huge amount of extra time.

Fair enough. In game, agreed.

 Peregrine wrote:

2) You're not in a game and are comparing two units. In this case comparing the average results for both units usually gives you enough information about their relative effectiveness, a more detailed analysis usually isn't going to add very much to the discussion.

As a statistician, I will tell you that a proper analysis will give you a much better assessment of what guns to take, especially when the number of guns is small. But whatever works for the individually really. I will admit, modelling things properly takes more work than most people are willing to put in. As easy as it is....

 Barksdale wrote:
As a statistician, I will tell you that a proper analysis will give you a much better assessment of what guns to take, especially when the number of guns is small. But whatever works for the individually really. I will admit, modelling things properly takes more work than most people are willing to put in. As easy as it is....

Could you give an example of a situation where you're comparing two units and just looking at the average outcomes tells you that A is better while doing a more detailed analysis tells you that B is better? And where the difference is significant, not just a case of a 0.1% difference either way where rounding error is more important than the type of analysis?

I think this example works for the question: An example where A>B, but B is actually better.

Ork boys T4 are more survivable than grots at twice the cost.
Grots T2 are much less survivable because most things wound them on a 2+.

A deeper analysis- Ork boys will be shot at more than grots because they pose more of a threat. On average boys get shot at 6 times more than grots making grots more survivable because there are less chances of them taking a wound than the boys.

Basically, no, because the second evaluation might use more accurate data.

Another analysis might say that Lascannons are better at killing tanks as opposed to autocannons. Then more data is added saying that 4 autocannons are better than 2 lascannons at killing tanks up to AV13.


Promised from earlier:

Standard deviation is a better guide because if you understand it, you start to realize that most of the time X will happen if you throw enough dakka at it. This is a common belief by orks who win the rng game by getting closer to the average because of their larger sample than other armies. You will also realize that the game is played 1 event at a time, so

The way I honestly play is that I know X should do about Y damage, then I realize that Murphy's Law is a stronger force (if it can go wrong it will, at the worst possible time). The next calculation I use is how much redundancy can I have to compensate for Murphy's Law. Luckily in 40k, the extreme cases are what people prepare for and remember due to their armies weaknesses.

Honestly I think simhammer.com's Resiliency-per-Point is the best way to look at ability to take damage against a given weapon.

Base resilience = 1/(((To Hit) * (To Wound) * (Fail Save) * (Fail FNP)) / (Wounds))

Divide this value by the cost of the model to get its resilience-per-point (RPP)

Base resilience per point = ((Base resilience)/(Point Cost)) * 100

This will give you an empirical method of comparison between units vs. a given weapon (normalized for cost). For example: A Plague Marine has an RPP of 224.9 against IG lasguns and an Ork Boy has an RPP of 100 vs. an IG lasgun.

But at high strength/low AP, an Ork actually beats out the Plague Marine. Look at a lascannon: Plague Marine: 11.2, Ork: 30.

It is common knowledge that the AC is better versus av 10 and and av11, when compared with the las. Right? Let us take a look.

First let us try using expected value operators to calculate the number of glances and pens we can expect for BS3 versus av10, for 1-5 heavy wepaons:

ac....glances.........pens.......las..glances..........pens
1......0.166667.......0.5..........1......0.083333......0.416667
2......0.333333.......1..............2.....0.166667.......0.833333
3......0.5..................1.5...........3.....0.25................1.25
4......0.666667.......2.............4......0.333333.......1.666667
5......0.833333.......2.5..........5......0.416667......2.083333

Well, as expected, the ac is putting out more glances and more pens each and every time. Surely the ac outperforms the las! Or does it?

Let us now instead look at the probability of destroying a 3HP av10 vehicle with a simple model using exactly the same probabilities per shot, just as above.

ac.....p(destroy)........las.....p(destroy)
1.....0.081597222....1.....0.138888889
2.....0.234073592....2.....0.258487654
3.....0.401593349....3.....0.391543175
4.....0.592360971....4.....0.525676684
5.....0.735171623....5.....0.646158417

Wait, whats that you say? The las is better than the ac versus av10 if you are taking 3 or less heavy weapons? Heresy!

What about av11? Let us look at the expected number of glances and pens, BS3, 3HP av11

ac....glances.........pens.....................las...glances..........pens
1......0.166667.......0.333333333......1......0.083333......0.333333333
2......0.333333.......0.666666667......2.....0.166667.......0.666666667
3......0.5..................1.............................3.....0.25................1
4......0.666667.......1.333333333.......4......0.333333.......1.333333333
5......0.833333.......1.666666667.......5......0.416667......1.25

Again, as expected, the ac is putting out more glances and pens.

Let us now instead look at the probability of destroying a 3HP av11 vehicle

ac.....p(destroy)........las.....p(destroy)
1.....0.054783951....1.....0.111111111
2.....0.14314641......2.....0.209876543
3.....0.263644264....3.....0.317704129
4.....0.404426782....4.....0.430618653
5.....0.539699944....5.....0.539690495

So, when taking 5 or less heavy weapons, the las actually outperforms the ac versus av11.

In fact, take a look at the probabilities. When taking only one heavy weapon, versus av 11, the las is over 100% better, for only a 16% increase in point cost when taken in a standard infantry squad (70/60). For two heavy weapons the las is 47% better. Three heavy weapons the las is 21% better. Four heavy weapons the las is 6.5% better. Five heavy weapons the las is just about identical, but can also deal with a ton of things that the ac cannot. For over 5 heavy weapons versus av11, weight of fire starts to have its impact and the ac becomes more effective.


Does it matter? Maybe not. But when the weapon slots are in short supply, and you are filling the holes in your list, it can help to have that little extra information to decide what weapons you should really be taking. Or not.

For RPP: Note that all of this is normalized by point, so while a Plague Marine is unquestionably better than an Ork boy stat-wise and equipment-wise, throwing your Ork allies against lascannons is better that using your Plague Marines against them. Not because Orks can take the hits better per-say, but because you will lose less effective strength per round of shooting against you when you throw the Orks against the lascannons.

The way i like to math hammer is to work out the 90 times out of 100 how many terminater that i will definitly kill.

Recall the out come of dices are binormial; but npq will give you variation. Sqroot it gives you sd. Mean -(1.96*sd) give you rough idea the minimum kill.

It is better to be sure... Very sure.

 Ixidor13 wrote:

Base resilience = 1/(((To Hit) * (To Wound) * (Fail Save) * (Fail FNP)) / (Wounds))

Divide this value by the cost of the model to get its resilience-per-point (RPP)

Base resilience per point = ((Base resilience)/(Point Cost)) * 100

This will give you an empirical method of comparison between units vs. a given weapon (normalized for cost). For example: A Plague Marine has an RPP of 224.9 against IG lasguns and an Ork Boy has an RPP of 100 vs. an IG lasgun.

Ork vs lasgun
1/ (1/2 * 1/3 * 5/6) = 36/5

36/5 / 6 = 6/5

6/5 * 100 = 120

120 != 100

http://ompldr.org/vaGxrZA/MATHS.jpg

Ohgod, I just noticed I had a flat "0.5" hit rate. This needs to be remedied!

Interesting discussion!

I think in most cases "normal" math hammer gets you as far as is useful given that you will be throwing only a handful of dice.

Distribution do matter sometimes. I have for example often been disappointed by mass lasgun fire vs tough targets and I just recently understood why.

As an example lets look at IG lasgun vs terminators.

For a single shot: p(kill) = (1/2) * (1/3) * (1/6) = (1/36)

Firing 36 shots the average outcome becomes 36 * (1/36) = 1.0 kills

However doing the math it turns out p(no kill in 36 shots) is actually about 36%, with a potential to do significantly more damage in rare cases.

The important lesson (for me at least) is that p (1+ kills in 36 shots) is actually as low as 64%, which is just slightly better odds than for passing a morale test at LD 7....

If you need that kill find someone else to do the job!



Math for those interested:

p(no kill in N shots) = p(miss) ^ N

p(miss) = 1 - p(kill) = (35/36)

p(no kill in 36 shots) = (35/36) ^ 36 = 0.363

As an addendum. Rolling hot in early turns or rolling bad in early turns has a magnified effect, as you or your opponent has less to work with in future turns. Timing is everything.

I think it would be interesting to get a probability calculator app.

Plug in your stats, plug in your opponent units stats and ask it the probability of killing x models or y models. THAT would be helpful as hell.

Need to take more stats courses before I can do that though......O well it will be fun distraction from that masters thesis.

Ok, sorry for the lack of detail earlier, I'll elaborate a bit on the methods used and interpreting data. As a disclaimer (I should have thrown that on the original post, but I think it got got cut off), the methods used are simply an expansion of the basic mathhammer formula.
For example, 1 wound on an Ork with Assault Cannon: 4 shots x (0.67 to hit) x (0.83 to wound) = 2.22 This is fairly basic, and how most people calculate it. I will note that this is in fact not a true probability, however it is the mathhammer approximation and gives us a reference for likelihood to compare to other numbers.

What I did was go a step further and set the desired outcome to more wounds.
For 2 wounds: You must roll at least a 3+ twice, and a 2+ twice to inflict 2 wounds. Thus, 2 wounds on an Ork with Assault Cannon: 4 shots x (0.67 to hit)^2 x (0.83 to wound)^2 = 1.24

Once again, it's a very basic approximation.

 greyknight12 wrote:
Once again, it's a very basic approximation.

No, it's a WRONG approximation. It is not possible to have a probability greater than 1.

I seem to end up being in this sort of situation quite often where this could be handy.

For example, in a Scouring mission, when there's different objective points, and two enemy units on them. Which one do I go for? I could wipe out and take the weaker unit on the lesser point. But, if I get a good roll, is that overkill? And could it potentially lose me the game by not going after the tougher unit on the 4 point objective.

In these cases, just caring about the average may end up costing me the game, whereas taking a gamble on the relative chances of the big hit, may win it with no chance of an opponent comeback.

I feel like I always have to mention binomial theorem at least once per mathhammer thread.

It is in fact, not possible to have a probability greater than 1 or 100%

This is why the average method is faulty, as it doesn't include variance, and in the case of multiple atttacks or multiple shots, gives you a misleading answer. (Like the odds of 8 lascannon shots killing a landraider...)

The above example about imperial guard killing a terminator is a perfect illustration.

A quick and dirty way that acts as an alternative to standard deviation is doing what I call the "known probability method". You can do calculations in a spreadsheet and then check your answers by using programming to roll 100,000 different examples and compare probabilties. It sounds crazy, but it's actually super simple.

You probably don't even realize it, but you do it all the time.
Snake eyes has a 1/36 chance of happening or a 2.77& chance of happening. Leadership 7 is passed (1+2+3+4+5+6)/36 = 21/36 or 58% of the time. Pretty much everybody knows these probabilities.

But other situations happen all the time too. A tactical squad rapid firing. A tactical squad charging. A tactical squad getting charged. Your landspeeder squad's alpha strike. Why wouldn't you just learn the probabilities for these common situations? You don't even have to memorize...you could just use index cards or on the back of your army list.

EDIT: Ailaros, remember we had that super huge discussion on game theory and I claimed that we aren't close to breakdown? Well let me point out....how many 40k players do you know that actually have a clue about their probabilities before they make a decision? Probably next to nobody does what I suggested. People still just do what they think is right intuitively, and that's still hugely exploitable, assuming equal armies.