Hi everyone, and fair warning: if you don’t like mathammer, you won’t want to read any further! (although the title alone should have scared off the timid)
This post is here to look at some more advanced mathammer and why it may be interesting to be aware of certain trends and how they may affect your games. I’m assuming a basic level of understanding of dice probabilities and I will try not to be too specific about certain situations, as the maths involved is NOT practical to try to use during a game.

I will present a few case studies, each of which should make a point which may influence how you think about your dice rolling. I don’t think I’m covering any really new ground and outside of the general principals, I won’t be going into too much actual maths (hooray!).
So, when we make a dice roll in 40k (or try to decide who to target based on our chance of wounding), I think most people have at least a vague idea of what we can expect from any given roll.
Some would go as far as working out the chance of each attack wounding and multiply that by the total number of attacks, to give the final expected ‘average’ number of wounds they are likely to inflict. This is generally a really good idea and is a helpful way of making snap calculations on the fly.

(chance to hit) x (chance to wound) x (chance to fail save) x (number of attacks) = number of expected wounds

I think that anyone willing to put that much thought into it wouldn’t be too disappointed if they didn’t get precisely the expected number of wounds, but sometimes (some would say often) the actual result doesn’t seem to conform to the expectation at all.
The truth is that if you add an extra dimension of analysis to the expected result, the chance of achieving the average number of wounds is actually lower than you might think.

‘Binomial distribution’ is a fancy way of saying that you work out the exact probability of getting that ‘average’ expected number of wounds, and shows you how reliable that estimate is.
In fact, you work out the probability of getting every possible outcome and perform an analysis of the resulting data.

The actual calculations are rather long-winded and boring (and are available to anyone willing to look it up), but by messing around with the numbers, you can get some interesting results.

Example 1: Tuna BBQ

So, a lucky Grey Knight Interceptor shunts up next to a conveniently positioned Tau Fire Warrior. He thinks this little fishy fellow is looking rather pale and generously offers to give him a light crisping with his Incinerator.
Hits automatically, wounds on a 2, no save.

Chance to overdo the spray-tan = 5/6 = 83.33%

No need to analyse further. We’ve got 2 possible outcomes and the fish is probably toast.

But what about the rest of the Tau’s unit? What if our hero can get a second warrior under his template?
Most people would agree that the most likely outcome is a double-grilling.
We expect to inflict an average of 1.67 (2 x 0.8333%) wounds, but as we know – you can’t inflict a fraction of a wound. It’s whole wounds only. If you calculate the probability of each possible outcome, you’ll see that:

You have a 69.4% (0.833 x 0.833) chance of getting both wounds,
A 27.8% chance of only getting one wound,
And a 2.8% chance of doing nothing.

So, even though it might seem like you are very likely to kill them both, 1 out of 3 times you’re only doing half as much damage.
What if the Tau have obligingly lined up, allowing you to fit 6 of them under your template?

You’d be right in thinking you’ll average 5 wounds, but you only have around a 40% chance of getting exactly 5 wounds.
You also have a 33.4% chance of getting 6 wounds, and the rest of the time (26.3%) you’re getting 4 or fewer.

Example 2: Minced Beef

A band of 30 Shoota Boyz ‘ave spotted sum ‘umies and are gonna fill ‘em fulla ‘oles.

With a 1\3 chance to hit, ½ chance to wound and 2/3 chance of the PA marines to save, 60 shoota shots are going to cause 3.33 wounds on average.
Following a binomial analysis of every possible outcome, we can find that those Boyz do indeed have the best chance of making 3 ‘oles at 22.5%.
However, they also have a virtually identical chance of getting 2 (19.8%) or 4 (18.9%) wounds. (incidentally, the chance of them getting 10 wounds from those 60 shots is about 0.12%)

If we strip the marines of their armour save, but keep everything else the same, the shootas are now averaging 10 wounds, but the result has become less reliable! A 13.4% chance to get exactly 10 wounds, and a roughly equal chance to get anywhere between 8 and 12 wounds. You’re NOT getting the average 86.6% of the time!

This is completely counter-intuitive! I was always sure that throwing more dice would make the average result more reliable – and if you were to throw a million dice and add up all the spots, you would get pretty reliably close to 350,000 every time. Unfortunately, that’s not what we’re doing, and so the same rules don’t apply.

Equally, increasing the chance of wounding decreases the reliability of the expected average! What?!?

Example 3: Chaos Kebab

Good ol’ Vindicare Assassin, he’ll set things right.
Hitting 11 times out of 12 and always wounding on a 2 or more, he’s sure to skewer an unfortunate devotee of chaos every time, right? Sort of.
Assuming no Feel No Pain or Invulnerable saves, he’s wounding 76.39% of the time. Over 5 rounds, he should average 3.82 wounds. It’s true that he’ll get exactly 4 wounds in 40% of his games, but 50% of the time he’s getting either 3 or 5 wounds.

If you were allowed to take two of these guys, between them they would get their expected average of 8 wounds only 29% of the time!

Example 4: Fruit Salad

Ok, let’s make up an extreme example. Snap shots, wounding on 6s, TDA, but you can have 200 shots to bring your expected average up to 0.93 wounds per round of shooting. Unfortunately, you now have a better chance to get ZERO wounds (39.5%) than ONE wound (36.8%) or more than one wound (23.7%).

You get the idea.

So, no-one can expect to carry out this kind of analysis during a game, and as we know well every situation is different, but there are a few helpful morsels to take away here.

-Good shooting breeds unreliability. The better your chance of scoring an unsaved wound (through good BS, high Strength or no saves), the more you are likely to deviate from your expected average.

-High volume shooting breeds unreliability. The more shots you take, the more you’ll deviate from your expected average. I guess this makes Orks even Orkier than most of us thought.

-Do you feel like you never roll as well as average? Hopefully this has started to explain why. Although you often have the chance to deviate and get MORE than the average.

-When you absolutely, positively gotta kill every mother fer in the unit (in one turn) you’re going to need a LOT more firepower. You know... to be sure.

10 BS4 Bolters vs T3 5+ models, a.Shots: 10 b.Hit Chance: 66.67% c.Hits: 6.667 d.Wound Chance: 66.67% e.Wounds: 4.444 f.Unsaved Wounds: 4.444 g.Models Killed: 4.444

Wounds:Chance: z.0 : 0.28% y.1 : 2.241% x.2 : 8.066% w.3 : 17.208% v.4 : 24.091% u.5 : 23.127% t. 6 : 15.418% s. 7 : 7.048% r.8 : 2.114% q. 9 : 0.376% p.10 : 0.03%

 Jancoran wrote:
Yeah Ive shown this math to a friend when he expressed some frustration over his Heldrake "Not doing enough".

 Scipio Africanus wrote:

 Jancoran wrote:
Yeah Ive shown this math to a friend when he expressed some frustration over his Heldrake "Not doing enough".

Shush. You don't use math.

OP, I like the fact that you included standard deviation. I'm just starting to get into using it myself along with a much brighter friend of mine, and I'm really finding it useful.

 Jancoran wrote:
...that is, if you let the units do what they are designed to do. So the trick in 40K isn't to know your odds. Its to know how to change them.

 Jancoran wrote:

 Scipio Africanus wrote:

 Jancoran wrote:
Yeah Ive shown this math to a friend when he expressed some frustration over his Heldrake "Not doing enough".

Shush. You don't use math.

OP, I like the fact that you included standard deviation. I'm just starting to get into using it myself along with a much brighter friend of mine, and I'm really finding it useful.

Hehehe. Oh I use math. My clients need me to be good at it.

But players who are obsessed that this or that didn't "work like it should" have one of two problems. either they are playing the wrong game due to an extreme lack of perspective or they are playing the wrong game because they are far too superstitious to be allowed around a random elemntal generor without ascribing some mysticism to it.

 Krellnus wrote:
The standard deviation is useful to be sure, but only for purely theoretical analyses, its too much trouble to calculate on the fly (for me at least) to have much use in mid game.

 Krellnus wrote:
The standard deviation is useful to be sure, but only for purely theoretical analyses, its too much trouble to calculate on the fly (for me at least) to have much use in mid game.

pocketcanoe wrote:
I can totally understand why people don't bother with this level of mathammer,

pocketcanoe wrote:
I can totally understand why people don't bother with this level of mathammer, as it took a maths teacher to explain it to me and the end result was really only that I now know WHY 'sh1t seems to happen more often than not'.

The quick probability calcs that most of us do are WAY more useful for WAY less work, but the binomial distribution only needs to be done once for it to really inform your decision making, and understand the reality of dice rolling, help to eliminate cognitive bias, etc.

Actually, cognitive bias in warhammer would also be a cool subject.


Automatically Appended Next Post:

 Krellnus wrote:
The standard deviation is useful to be sure, but only for purely theoretical analyses, its too much trouble to calculate on the fly (for me at least) to have much use in mid game.

Exactly. Just an awareness of the general trends is all you need in the back of your mind could help you avoid one or two duff decisions in a game.

Actually, one other thing I've taken from this is that it has put me off the idea of a hoarde army, as this makes them far less tactical and far more reliant on 'good' dice rolls than I previously thought. So I guess it's as useful in the army selection/list building process as it is in-game.

 Scipio Africanus wrote:


Weight of dice will do anything, anyone with any idea of math should know that.

Scipio Africanus wrote:

 Krellnus wrote:
The standard deviation is useful to be sure, but only for purely theoretical analyses, its too much trouble to calculate on the fly (for me at least) to have much use in mid game.

Got an excel capable tablet?

I've got a program for you.

pocketcanoe wrote:I can totally understand why people don't bother with this level of mathammer, as it took a maths teacher to explain it to me and the end result was really only that I now know WHY 'sh1t seems to happen more often than not'.

The quick probability calcs that most of us do are WAY more useful for WAY less work, but the binomial distribution only needs to be done once for it to really inform your decision making, and understand the reality of dice rolling, help to eliminate cognitive bias, etc.

Actually, cognitive bias in warhammer would also be a cool subject.


Automatically Appended Next Post:

 Krellnus wrote:
The standard deviation is useful to be sure, but only for purely theoretical analyses, its too much trouble to calculate on the fly (for me at least) to have much use in mid game.

Exactly. Just an awareness of the general trends is all you need in the back of your mind could help you avoid one or two duff decisions in a game.

Actually, one other thing I've taken from this is that it has put me off the idea of a hoarde army, as this makes them far less tactical and far more reliant on 'good' dice rolls than I previously thought. So I guess it's as useful in the army selection/list building process as it is in-game.

pocketcanoe wrote:
Weight of dice does NOT always guarantee results!

 Krellnus wrote:


I've just got my iPad with Excel on it, but to be blunt my point was thus, if it gets to the point where you're in game and two alternatives are close enough that you need to get out pen & paper or a tablet, or a laptop or even a binomial probability to make your decision and you can get away with it, your opponents would have to be very accommodating. Yes, a laptop can speed up the process, but it is still time consuming.

Your best bet would be to memorise by rote a list of generic scenarios you would encounter, the results for which you memorise, still quite a task imo.

 Krellnus wrote:
Numbers? I take it that is some program with a similar purpose to excel? Unless you mean do I actually have some numbers available?

Let me pose it to you like this.

If you are playing a game with someone and they doing maths calculations/putting numbers into a spreadsheet, even if it doesn't take long, it is no less disrespectful to your opponent than sending text messages to your mate during a game.

 Scipio Africanus wrote:

pocketcanoe wrote:
Weight of dice does NOT always guarantee results!

Yes it does.

You're saying it doesn't guarantee that we get the same results every time. I know that. I don't need to be told that.

What I'm saying is that you're wrong to believe that they won't get the job done. They may not get the job done consistently, but they will get the job done.

For example 150 BS3 twin-linked lasguns will do anywhere from 0 to 10.2 wounds to T6 with a 3+ and 5+ fnp. (a standard deviation of 2.01 from an average of 4.17) That doesn't mean I can't hinge on the dice doing anywhere from 2.15 to 6.18 wounds, or one standard deviation. 67% of the time, that's what will happen.

I think it's useful when you have a small number of attacks/shots/whatever, say 3, and the average might be, oh I dunno, 30% per shot, and you want to know the chances of getting 0 successes, 1 success, 2 successes or 3 successes. You might know the AVERAGE is about 1, but knowing the actual distribution makes it easier to make an informed decision.

It's also useful for determining which is better, a large number of attacks with a low chance of success, or a small number of attacks with a high chance of success but the same average.

Scipio Africanus wrote:

 Krellnus wrote:
Numbers? I take it that is some program with a similar purpose to excel? Unless you mean do I actually have some numbers available?

Let me pose it to you like this.

If you are playing a game with someone and they doing maths calculations/putting numbers into a spreadsheet, even if it doesn't take long, it is no less disrespectful to your opponent than sending text messages to your mate during a game.

How does one have excel on an iPad? I was unaware microsoft had released it there.

Numbers is Apple's more aesthetically pleasing but ultimately weaker version of Excel.

I disagree with that. If my opponent asks me not to, I won't, but it's not like I'm detaching from the game during his turn, I'm still in the game, I'm just doing something to do with my half of the game during the time I can't do anything during his.

also, what do you mean by "sending text messages" during a game, do you mean the kind of messages that are sent in conversation? I wouldn't have a text conversaiton during a game, but answering a text or an important phonecall doesn't really register to me as rude. (of course, you'd excuse yourself if you had to answer the phone, with apologies.)

pocketcanoe wrote:
When did I say I believe it can't get the job done?

Weight of dice does NOT always guarantee results!

It has a good chance of MORE than getting the job done. It also has a roughly equal chance of getting less than the job done, in addition to a reasonable-to-low chance of getting EXACTLY what you need (depending on specifics).

Overall, rolling more dice will improve your chance of achieving (or achieving more than) a fixed target roll. That's a fact.
Clearly, I'm more likely to get 6 or more sixes on 36 dice than I am on 6 dice.

Also, rolling more dice will DECREASE your chance of of getting EXACTLY the average number of your target roll.
I'm more likely to to get exactly the average of 1 six on 6 dice than I am likely to get exactly the average of 6 sixes on 36 dice. That's more counter-intuitive, but it's still a fact. That's all I'm sayin'!

It's only in that sense that rolling more dice becomes less reliable. It doesn't affect your average, but it does affect your chance of getting exactly the expected average. Because you're more likely to deviate from the average, you're more likely to roll worse. You're also more likely to roll better.

The difference between 2.15 and 6.18 wounds is a huge variation for a small number of wounds! And we're only getting within that range 67% of the time? That's an unreliable result (in my opinion). If I NEEDED to cause 4 wounds (the approx. average) and had 150 rolls to get it, I would intuitively be annoyed if I didn't get my 4 wounds. The truth is, I have a really good chance of not getting what I needed. Also a really good chance of getting more than I needed.

I think we're arguing different points, TBH.

Krellnus wrote:
They recently released it as part of Office 365, without a 365 subscription, its a free reader of the relevent formats, with it, it is a fully fledged tablet version of the program, only came out last month or so irrc, one of my lecturers told me about it last month anyway.

For me (and many people I play with) it would be considered incredibly rude to stop paying attention to the game and start doing something, because you are basically saying "after setting this time to play with, I have something else I want to do now so I'm going to stop paying attention".

Krellnus wrote:I don't think anyone was insisting on expecting the exact average.

 Scipio Africanus wrote:

If I say 150 lasguns should do so many wounds, but could do M-SD through M+SD, I plan for them doing M-SD. What's the problem with this?

 Scipio Africanus wrote:

Krellnus wrote:
They recently released it as part of Office 365, without a 365 subscription, its a free reader of the relevent formats, with it, it is a fully fledged tablet version of the program, only came out last month or so irrc, one of my lecturers told me about it last month anyway.

For me (and many people I play with) it would be considered incredibly rude to stop paying attention to the game and start doing something, because you are basically saying "after setting this time to play with, I have something else I want to do now so I'm going to stop paying attention".

Ah, Okay. I was unaware.

I disagree with the second statement, but that's a personal thing. If I played you, and you asked me to stop doing something else I Would, because you've made it clear that it upsets you. Doing it after that point is undeniably rude. But, if I get a text message - I don't have text conversations, so that's not an issue you'd experience against me, I will respond to it during an opponent's movement phase, if they don't need my assistance after I've offered it. I guess I just don't feel the same way you do about text messaging. All that said, ideally I'd answer a text message quickly so I don't waste play time. Of course, I dislike someone who has a conversation with a person in the flesh during my turn, especially the assault phase which really needs both players paying attention, so I think I understand your qualm.

 Scipio Africanus wrote:

Krellnus wrote:I don't think anyone was insisting on expecting the exact average.

I think OP is kind-of assuming we're always aiming for exact averages, which is never the case.

Overall, rolling more dice will improve your chance of achieving (or achieving more than) a fixed target roll. That's a fact.
Clearly, I'm more likely to get 6 or more sixes on 36 dice than I am on 6 dice.

So, we aren't arguing here.

Also, rolling more dice will DECREASE your chance of of getting EXACTLY the average number of your target roll.
I'm more likely to to get exactly the average of 1 six on 6 dice than I am likely to get exactly the average of 6 sixes on 36 dice. That's more counter-intuitive, but it's still a fact. That's all I'm sayin'!

Cool, I'm not ARGUING with this. I KNOW this. It's not COUNTER INTUITIVE. It's Larger numbers = larger variation.

 Scipio Africanus wrote:

 Jancoran wrote:
...that is, if you let the units do what they are designed to do. So the trick in 40K isn't to know your odds. Its to know how to change them.

For the trick of 40k to be knowing how you change your odds, inherently, you must know your odds.