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Seriously, feel free to laugh but I'm a lawyer by trade. Haven't mathed in years. I know that working out the math is pretty important for competitive list and I'd really like to get more competitive but thinking about it makes my brain shut down. If anyone can just show me the basic math for how to calculate wounds, rerolls, and all that it would be greatly appreciated. I honestly don't even know where to start.

The internet is always available to math for you:

http://www.mathhammer8thed.com/

Should get you moving in the right direction.

I didn't know this existed. Thank you so much!

It is all about probabilities and knowing how to use fractions.

First up, the easiest way to calculate a probabilits is to take the number of benefitial outcomes and divide them by all outcomes.
for example, a Marine hits on a 3+, which means 3, 4, 5 and 6,which is the benefitial outcome. How many possible outcomes are there? 1, 2, 3, 4, 5, 6. That means a Marine has a chance to hit of 4 divided by 6, also known as 4/6, which can also be written as 2/3.

Next you need to know how to multiply and add fractions. Say a Marine wounds a target on a 4+, which means 3/6, which is 1/2. Now you have two fractions. Multiplying fractions is simple. Top number times the top number and bottom number times the bottom number. This case it is 2/3 * 1/2, meaning 2*1 / 3*2, which 2/6, which is 1/3. So a Marine has a chance of 1/3, meaning 33.33333333333% to wound a T4 target with a single Bolter shot.

Adding fractions a bit tougher. Say you have a scenario, where a Marine re-rolls to hit. So he hits 2/3 of the time, which also means 1/3 are misses. Those misses can be re-rolled. The equation would be 2/3 + (1/3 * 2/3). The brackets are important as the multiplication needs to happen before the addition. What does the bracket mean? Well, for every miss, which is 1/3, you have a chance to get another hit, 2/3. For bracket is added, because those re-rolls come in addition to the hits you made.
Adding 1/3 and 2/3 is simple. You just add the top numbers while leaving the bottom ones the same. The result is 3/3, which is 1. This only works if the bottom numbers are equal. Say you need to add 2/5 and 3/10. How do I make the bottom numbers equal? Well, you can multiply 5 by 2, which gets you 10 at the bottom. Since you multiplied the bottom number by 2, you need to do the same with the top numbers. So 2/5 equals 4/10, which can be added to 3/10, which is 7/10.
Edit: The example above would be 2/3 + 2/9, which is 6/9 + 2/9, which is 8/9.

Well, this is a very basic overview. What you need is to know how to calculate fractions and probabilities. Have a look at a probability tree, which is the essence of mathhammer. Google will be your friend.

If you want to take it to an extreme, you can also have a look at condition probabilities or even statistical distributions, which I like to use for cummulative probabilities to find out the exact chance of killing something, including overkill.

I did this in statistics when I went to uni, so this is second nature to me, but the basic stuff is easy to learn. If you want, you cam PM me and I'll give you some examples to practice with

It's always good to know the reasons behind the math, as they let you do some mental gymnastics on the fly. You don't even need to calculate much, just work in how many "points of failure" you have, and you can get a good feel on whether your actions are going to have rewarding enough consequences.

Math Lesson #1 - Each additional stuff is multiplication
So, a 4+ to hit is a 50% (aka 1/2 = 0.50). A 4+ to wound is also 50% chance to wound (aka 1/2 = 0.50). So to hit AND wound, you multiply them together. 0.5 x 0.5 = 0.25 (aka 1/2 * 1/2 = 1/4). So that means that "some, but not a lot" are going to hit and wound. If they have a 4+ save on top of that, that's another 1/2 of the remaining results getting through, meaning a small amount. Start changing around those numbers, and you'll see how high or low you go.

Math Lesson #2 - Rerolls add back in a portion of the "failed" rolls
So, if you have a 4+ to hit, but you get to reroll failed hits, that means you hit 1/2 of the time, and miss 1/2 of the time, but then you get to reroll those 1/2 misses for another 1/2 hit and 1/2 miss. This means you get 1/2 + 1/2 of 1/2 = 3/4 (aka 0.5 + 0.5*0.5 = 0.75). If you only reroll 1's in that case, you'll only reroll a third of those misses (1, 2, or 3 are the possible outcomes of a missed shot, and you only reroll 1's, so that's 1 of the 3 of them, so that's 1/3 or one third). Remember that only half of those rerolled 1's are going to hit, so 1/3 * 1/2 = 1/6. Add that to your original hits, and there you go (1/2 + 1/6 = 2/3).

Math Lesson #3 - All of these numbers are "50% of the time", meaning that big skews can happen, but usually the worst or best you'll face is half or half+half the number you expect
The average result of 2d6 is 7, but that still only happens 1/6th of the time. The rest of the time, it's something other than 7. But 7 is still the most common, and the average of a bunch of dice rolls of 2d6 will be 7, so 7 is a good one to go with. Same applies to 40k math-hammer; any probability may be the most likely, and you'll often be around that number, but actually getting that number exactly should NOT be expected. Thankfully, the further you get from that average number, the less likely that result becomes.

As such, if you figure you should kill 5 guys from shooting, you can reliably expect to get half that more or less, so between 3 and 7 guys will actually be taken out by the shooting (5 divided by 2 is 2.5, so 2 higher or lower). Even then, there's rolling hotter or colder than this, but if you're happy with that "3" result, then you should feel good going for it. If you take another 1/2 of that and apply it again and you can start being reasonably confident (2.5 divided by 2 is 1.25, add that to 2.5 for 3.75, and that's your variability off of 5, so between 1 and 9 guys).


Those should help!

Yeah, I must admit, I've never been incredibly good at expressing stats in decimal form when it comes to 40k mathhammer. I always have to do the math in my head using increments of 6.

For example, 6 rapid-firing bolters should hit 8x and would 4x against T4 models.
If the mathhammer is not divisible be 6, or can be halved, I rely too much on rounding.

On the bright side, this has lead me to often do the math in a way that seems to account for "bad dice" and thus I am rarely in a poor position.

-

It's all stats. Learn stats.

As for compensating for bad dice, use the stats to get the numbers at the confidence level you want, or use the math to get the probability that they do that well or better.

Basically, if you want to be better at math, read math books.

If you want to be better at Warhammer, it's not been conclusively shown that mathhammer skills are the way to do it.

You get Mathhammer on this forum showing some obviously BS things. Like 6 STrike squads successfully charging 6 Razorbacks after deepstriking. Or Reapers removing 10man Tac Squads every time.

Also don't try to factor in multiple turns. Your chances are no longer independent so correct math becomes A LOT more complex.

The thing is, averages do not represent the system very well. An average of 3.5 on a D6 does not mean it will happen all the time, just that if you sum up all D6 roll up to infinity and divide by infinity, you will have an average of 3.5. The correct way to calculate would be the Bernoulli-Laplace chain (aka binomial distribution) and use cummulative probabilities. In simple terms, the chance of rolling a 4 is 1/6, same for 3, 2 and 1, so rolling a 4 or less is 2/3. You need to that calculate the overall chance that, for example, 3 Lascannons can kill one Rhino, taking into account hits, wounds, armour saves and different damage rolls, including overkill and adding them all together (cummulative probability). That would be the most precise way of doing mathhammer... but that's too time consuming, so tend to stick with averages and intuition

Fortunately there is a proper mathammer calculator for this edition which does exactly that.

mathhammer.thefieldsofblood.com

 Yarium wrote:
If you only reroll 1's in that case, you'll only reroll a third of those misses (1, 2, or 3 are the possible outcomes of a missed shot, and you only reroll 1's, so that's 1 of the 3 of them, so that's 1/3 or one third). Remember that only half of those rerolled 1's are going to hit, so 1/3 * 1/2 = 1/6. Add that to your original hits, and there you go (1/2 + 1/6 = 2/3).

Nah, it's 1/2 + 1/2*1/6 = 7/12 = 0.58333.

You don't take the 1 out of 3 misses, you consider the whole 6 outcomes, 3 of the outcomes are hits, 2 of them are misses, 1 of them is a reroll, so before simplifying the fractions it's 3/6 + 1/6*3/6.


Automatically Appended Next Post:

 Trade_Prince wrote:
The thing is, averages do not represent the system very well. An average of 3.5 on a D6 does not mean it will happen all the time, just that if you sum up all D6 roll up to infinity and divide by infinity, you will have an average of 3.5. The correct way to calculate would be the Bernoulli-Laplace chain (aka binomial distribution) and use cummulative probabilities. In simple terms, the chance of rolling a 4 is 1/6, same for 3, 2 and 1, so rolling a 4 or less is 2/3. You need to that calculate the overall chance that, for example, 3 Lascannons can kill one Rhino, taking into account hits, wounds, armour saves and different damage rolls, including overkill and adding them all together (cummulative probability). That would be the most precise way of doing mathhammer... but that's too time consuming, so tend to stick with averages and intuition

To use binomial distributions you need to figure out the chance of 1 success, which you need to do for calculating the average anyway.

Once you have that chance of 1 success you can just plug it in to a binomial calculator like this one...
http://stattrek.com/online-calculator/binomial.aspx

The best advice I, or anyone can probably give in regards to mathhammer, is use it as a tool to help make decisions in list building and in game, but do not let it make the decisions alone for you.

It can be great for working out certain values in list building – such as points per wound ratios etc, but, always take into account your list synergy and ideas as well.

In game, it’s a great tool for helping with target priority for certain units and trying to work out the best tool for the job.

You can easily calculate most basic mathhammer with your phone by memorising 5 numbers.

Let's say you've got X shots and you want to know the number of hits you'll statistically get. Just look at the BS.

2+ means multiply by 0.83
3+ means multiply by 0.66
4+ means multiply by 0.5
5+ meand multiply by 0.33
6+ means multiply by 0.16

That's all there is to it, same with to wound and saves.

Let's take you through an example: You've got 10 shots. First, you roll to hit: multiply 10 by 0,66. The result is 6.6. Which means that, on average, you're looking at 6 to 7 hits.

Next you calculate your to wound roll which we'll say is 4+. You take 6.6 and multiply by 0,5. The result is 3.3 which means that on average you'll get about 3 to 4 wounds.

Next the enemy gets to save and the enemy has a 5+. Now you have 2 options:
- You multiply 3.3 by 0.33, you get 1.1 which is the number of SUCCESFUL saves. Substract this from the amount of saves they made (so 3,3 - 1,1) and you've got the number of unsaved wounds, 2,2 in this case.

- You can do it quicker by using the multiplier of the opposite end of the list. If their save is a 5+, you could use the 3+ multiplier to instantly calculate the number of unsaved wounds. So instead of using 0,33 (the 5+ multiplier) you use 0,66. If you do 3.3 multiplied by 0,66 you instantly get the 2.2 UNSAVED wounds you're looking for.

So to calculate unsaved wounds the list looks like this:
2+ save means multiply by 0.16
3+ save means multiply by 0.33
4+ save means multiply by 0.5
5+ save means multiply by 0.66
6+ save means multiply by 0.83

When I was at school I could never explain how I got to my answer, but somehow always got there. Similar with 40k I work out roughly in my head the chances of success before deciding on my course of action.
as above it can be broken down and explained, but frankly this just confuses me.

For more complex probability estimation, use Bernoulli trial

The easiest way to give a rough idea or compare units is using averages. All you need to know are the probabilities of rolling numbers on a D6:-

2+ (5/6 or 'five sixths')
3+ (4/6 or 'two thirds')
4+ (3/6 or 'half')
5+ (2/6 or 'one third')
6+ (1/6 or 'one sixth')

So if your unit has 18 shots requiring 3+ to hit and 4+ to wound, you would calculate: two thirds of 18 is 12 hits, half of 12 is 6 wounds. If the enemy unit had a 3+ save then two thirds of 6 is 4 saves, leaving 2 wounds/damage (on average).

It takes a bit of basic maths but the more you do it the more it starts to come naturally.

I really appreciate all the help everyone. I didn't expect to get this many responses! I mainly wanted to use it for list building to get a more realistic expectation of the damage a unit will do given any given target. Especially when I'm looking at building a new army (I'm doing that now). Having it broken down like this barney style doesn't make it seem so bad. Thanks everyone!

1d4chan's mathhammer article.

For those opposed to percentages I recommend the 216 method which does everything with integer multiplication:

Multiply the good outcomes of the dice you roll to hit - 3 in the case of 4+ by the number of good rolls to wound , by the number of failed rolls for armour save. e.g. a Tactical marine boltgun at guardsmen would be 4 possible hits, by 4 possible wounds, by 4 possible failed saves gives 64. Write it as the fraction of 64/216 and you can see modern boltguns just arent that good

HBolters on the other hand (4*4*5) are very good, plasma guns (4 * 5 * 6) extra very good - but not as much as might be expected. Sometimes you don't even need the whole 216 - you can just tell by the 3 numbers multiplied that weapon A is so much better than weapon B, or conversely only a little better as is the case with the HBolter and PGun.

DaPino wrote:
You can easily calculate most basic mathhammer with your phone by memorising 5 numbers.

Let's say you've got X shots and you want to know the number of hits you'll statistically get. Just look at the BS.

2+ means multiply by 0.83
3+ means multiply by 0.66
4+ means multiply by 0.5
5+ meand multiply by 0.33
6+ means multiply by 0.16

If you have a phone to calculate the maths, just keep them in sixths, then the only number you have to remember is 6 - the number sides of the die you're rolling

Count the number of "successful" outcomes, and that's your ratio. Need a 4+ to hit? That's 3/6 because you have 3 outcomes that are successful. Need a 5+ to wound? That's 2/6 because 2 outcomes are successful. You opponent has an armour save of 3+? That's 2/6 because 2 outcomes result in a success for you (them rolling either a 1 or a 2).

So the chance is 3/6 * 2/6 * 2/6. Multiply that by the number of "attempts" to get the average, so 10 shots is 3/6 * 2/6 * 2/6 * 10 = 0.55, thus on average you kill just over half a dude for each round of shooting

People who have been calculating stats for a while will tend to simplify the sixths (ie. 3/6 = 1/2 = 0.5) but that creates an extra hurdle for the mathematically challenged to understand what they're actually doing.

also remember to reduce the 'expected' outcome on important rolls by about 33% and vice-versa for unimportant ones, dice are spiteful dreamcrushing creatures

Don't sweat it.

You don't need the exact figures. Just an estimate of how likely you are to kill something.

When you are playing an actual game, you won't have the time do the calculations anyway.

Also? Any opponent worth their salt will make their best to alter your equations on the fly.

And what might feel like an overkill might just be what's needed to do the job.