What Is Math-Hammer?
I'm going to use Math-hammer as a term to describe the rudimentary application of statistics to 40k in an attempt to figure out the outcome of a discrete round of harming the other guy.
Math-hammer usually takes the form of something like this:
Number of shots * chance to hit * chance to wound * failed saves = Number of casualties.
Let's put this neat little model into action: I fire my 10 Dire Avengers into your Imperial Guardsmen that are in cover. The pertinent facts are:
1) Dire Avengers are BS4
2) Their guns are 2-shot, S4, AP5 jobs
3) Guardsmen have T3, and a 4+ cover save.
Let's turn that crank with the following data:
Number of shots = (10*2) 20
Chance to hit (at BS4) = 2/3
Chance to wound (S4 vs T3) = 2/3
Failed saves (4+ cover) = 1/2
Ergo: 20 * (2/3) * (2/3) * (1/2) =...
20*(2/3) = 13.333 hits * (2/3) = 8.8886 wounds * (1/2) = 4.44 Casualties.
Hooray! We just killed some Guardsmen!
So, what's the problem with this model? There are a few: namely, the dice, the laws of statistics, and template weapons.
Dice are not necessarily impartial random-number generators. Some people suggest that Chessex dice (the default, little rounded-corner boggarts) lean towards the 1 a little bit. There are also some nasty tricks you can do to manipulate dice. My personal bias involves just playing with the Chessex dice, and the hell with the bad rolls you get. Actively trying to rig your rolls is a punk thing to do, and should realistically get you kicked in the dice bag.
The bottom line is that the dice do not always behave in a perfectly predictable manner, but we can still make some basic assumptions about how they'll act any given time we roll them.
Laws of Statistics
Think about most the times you roll dice in 40k. Maybe it's an ordnance shot or blast weapon; then it's 2d6 and the scatter dice (...just TRY running some math-hammer for scatter. It's pretty rough). Maybe it's your Predator Destructor kicking out 8 shots in a turn. Maybe it's your tactical squad rapid-firing someone with about 16ish bolt-weapon shots. Maybe it's terminators on the charge with 15 attacks.
What do a lot of these have in common? It's not a lot of dice. Statistics works best with large samples; a good statistical sample has a minimum of 30ish trials. Some units can indeed pull that off in a single action (IE: Orks shooting/charging, Fiends of Slaanesh in assault, 30-man guard squad w/ First Rank Fire, Second Rank Fire), but most cannot. Some units cannot throw thirty dice in the course of the game, even if all seven turns happen.
What's that mean for you? Even through the course of a game, you cannot necessarily expect 'average' performance out of a given unit.
Now, if we take a basic weapon that has fixed chances of hitting, we can run the rest of our formula from there, yes? Same goes for assault. The problem lies in predicting how many hits you can slap on a target with a flamer or blast weapon. Templates have to take into account how close models are, and the formation they are arranged in. If everyone is standing shoulder to shoulder with a 2" max coherency spread, it's a much different picture for blast weapons than if they're all in base-to-base contact with each other like a fresh Deep Strike formation, for example.
The best we can do for Blast Weapons is model their scatter, armed with the knowledge that 2d6 averages about a 7, and there's a 33% chance of getting a 'hit' result on the scatter dice. So, right off the bat we know a blast weapon will hit 33% of the time. 33% of the time, scatter will then average 7-BS, so we have some idea how far the center dot might go. About the only place we can use this knowledge is when we try to apply it to vehicles and how big their hull is; we have a better chance of landing a blast template on a Land Raider than a Killa Kan, for example.
Why use Math-hammer, then?
Why, indeed? Because it gives us a starting point for making decisions. Common sense tells us that one Guardsman is unlikely to beat up a thunder hammer storm shield terminator, but it IS still possible. On the other hand, some things get a little closer. Math hammer gives us tools to make a guess at the outcome of shooting or assault, and that's at least something to base your decision on.
The numbers are not always accurate. However, it's a darn sight better than just using wishful thinking and sacrifices to the 'dice gods' as a means of making decisions with your army's firepower.
Using Math-Hammer intelligently
Here are some suggestions for using Math-hammer to your advantage:
1) Know the limits of the stats
-You need 30 rolls to get an accurate 'sample size'; if it's below that than you're not really guaranteed an 'average' result
2) Know the limits of the dice
-Dice are not perfect random-number generators, either through manufacturing processes or human interference a la practiced rolling.
3) It's Still Math
-Math, when done right and mindful of its problems, is STILL a reasonably reliable when you follow the steps.
-At least, math is a reliable tool given an understanding of the tool. It's like a computer: it can only do what you tell it to do, how you tell it to do.
4) Know what the 'Average' means
-The 'average' result is the midway point, which means that IF you roll enough dice, THIS is what happens about 50% of the time.
-What you're really dealing with is a Normal Distribution (aka bell-shaped curve)*. You're hoping that your dice will land somewhere in the middle, but it is still entirely possible to see a given set of rolls fall way the hell away from average (hey, it's possible to see 5 1's, [0.01% chance, theoretically] and you have the same chance of seeing five 6's in a row as well)
5) Know that Things Can Still Go Wrong
-The dice are still fully capable of throwing complete outliers at you. We call this 'the wheels coming off', when you fire five bolters into a single guardsman in the open, and he still lives, for example. (Worse still, he kills 2 guys on the charge, and lives.)
-In other words, you care about math, but math doesn't give a flip about you, and can screw either of you over, or both of you, or neither of you. Crap happens. Deal with it.
6) Realize the limits of Math-Hammer
There are things Math-hammer simply will not do, and they are:
-Tell you WHERE to move
-Tell you what your opponent will do
-Tell you how many of your infantry will actually be in range of your target
-Tell you a blessed thing about how many people you actually will hit with a template-based weapon
Math-Hammer CAN give you an idea on:
-How much fire you need to kill a target
-How much fire you need to force a morale check
-Whether or not you might make assault (remember, you just need base-to-base with ONE guy)
-The chances of immobilizing yourself in terrain
-The chances of you winning/losing an assault, and potential morale results
Math-hammer is a tool. No more, no less. It's not perfect, but it's a workable enough basis for making decisions about dice in the game. As such, I suggest a working knowledge of it, since this IS a dice game, and Math-hammer DOES tell us a little how the dice work. It's a more reliable basis than all the little pre-concieved notions that your brain tends to have.
Any kind of theory makes assumptions. The more you know about the assumptions, the more use you can get out of the theory. At the end of the day, theory itself is a tool. Theory simplifies the world, loses some resolution, and usually gives you a (reasonably reliable) method of making decisions and/or learning about an aspect of life, the universe, and everything.
Math-hammer just gives you a rough idea of what can happen when you throw the dice, and that's better than nothing, in my book. There are days where the average will happen every single roll, there are days where the dice will roll sub-par on a game level for you, but in the end, across enough games, the dice will roll average.**
*Disclamer: I'm being a little lazy here and linking to Wikipedia. Note that since Wikipedia is open to anyone who wants to edit it, anyone can (and in some cases, has) tweaked the data in an incorrect manner. If you really, truly want solid info, wikipedia is a good start, but then you'd better check out the sources they've listed in the article, and move from there.
**Unless you're a bloody dice cheat. Or the other guy is.