Back to Basics: Mathammer

I was kindly sent this e-mail about Mathammer and I thought it was a great way to start out the Back-to-Basics series. So big thanks to Dariokan for writing this up!

Hey Kirb !

I’ve read quite a lot of crap on the Internet recently (including once on your nervertheless excellent website, it was about a Dread That so wasn't me... though I do like how 3++ was complimented haha) and I told to myself: “Why not sharing a little knowledge about mathammering, and contributing a little”. Then I got drunk and forgot. Most of players are already aware of those, but here are some quick tips for those who are not that keen on math, to help calculate probabilities and odds. And excuse my English; it’s not my first language.

1) Probability ? What the @#+*?

It’s the (relative) frequency that an event will occur. Let’s take Guardsmen and their very convenient CT3 as an example. They hit on a 4+, that means if the die reads 1, 2, 3 the Guard misses, and 4, 5, 6 he hits. The event we are talking about here is when the Guard will hit. The die can show us 6 results, 3 of which lead us to the event “hit”. Therefore, a Guard has 3/6, same as ½ aka 50% chance to hit. And 50% chance to miss. A Space Marine will hit on a 3+, that’s 3, 4, 5 or 6 on the die. 4 results that will lead to the event “hit”: 4/6, or 2/3 that means roughly 66% to hit. Of course, that’s does NOT mean that if you shoot with 2 Guards, one will miss, and one will hit. We’ll discuss about it later.

2) Combining probability

“Yeah right, but you roll to shoot, then you roll to wound, then there is an armor save." And that’s why we must combine numbers. Following the same pattern as above, you will have to multiply the numbers to get an estimation of the chances that the event “The target loses one hp” happens. Let’s take a closer look at it. Keeping with a Guard shooting at another Guard.
-To hit: 4+, that are 3 chances out of 6, 3/6, or ½.
-To wound: 4+, same: ½
-NOT to save: a Guard as a Flak Jacket, giving him a 5+ save. We are looking for the event “the wound is not saved”, which means results of 1, 2, 3 or 4 on a die. 4 chances out of 6, 4/6=2/3
-Final result: 1/2x1/2x2/3=(1x1x2)/(2x2x3)=2/12=1/6
When shooting a single shot at another Guard, a guard as 1 chance out of 6 (roughly 16%) to kill him.

Of course, those calculations can be applied to vehicles penetration, and so on.

3) Cranking probabilities up to 100%

“Does that mean if I shoot with 6 guards, I then have 100% chances to kill him?” No. Nononono, absolutely not, just go hang yourself in a tree if you’ve ever thought that. Simply put, apart from automatic rolls (i.e. rolling to see if I penetrate a AV10 vehicle with my S10 weapon, meaning that any result will do) you can never be 100% sure that something will happen. To have the correct numbers, you must go through a “Bernoulli trial”, which is complicated and time consuming. And not the purpose of this text. But math education...:(

Off course, the same applies to automatic fails. Penetrating a Land Raider with a Lasgun is impossible, but wounding a MC with E6, a 2+ save and a second 2++ Super Fell No Pain From Hell is still possible. If highly unlikely. (0,42% for those who wonder. 1 out of 216)

4) Law of large numbers

Long story short: if you throw 2 dice, you will have very random results. If you throw 30 or more dice, you will get closer to the expected result. Eluding numbers, because I do not want to complicate things, if your 2 Guards shoot each once, the odds to hit or miss are like that:
25%: no one will hit.
50%: only one guard will hit.
25%: the 2 of them will hit.

On the other hand, if you throw a bucket of dice (First Line FIRE! Second Line FIRE! Or a mob of 25 boyz charging), you can be sure that the results will get closer to the expectations. That means, for 10 guards firing 3 times, 30 shoots at 4+, most of the time you will be close to 15 hits. 12, 14, 18, but almost never the results “no one will hit” or “all of them will hit”. That’s the main goal of concentrating fire.

5) Estimations on the run

So you want to know what will probably happen when your unit will shoot at this other unit? There’s a simply way to know, on average, what is going to happen. Let’s imagine 10 Guard rapid-firing at another squad.
Base pool : 20 dice. 50% will hit -> 10 dice remaining.
10 dice to wound, again 50% chance to wound -> 5 dice remaining.
Armor save : 4 wounds out of 6 will not be saved : 4/6=2/3. What is two third of 5 dice? 5x2/3=10/3=3.33. That means, on average, you will kill between 3 and 4 guards.
Food for thought : what now if they shoot at 10 marines ?
20, shoot at 4+: 10 remaining. 10, wounding on 5+: 3.33 dice remaining. Armor save 3+, means only 2 wounds out of 6 (1/3) will not be saved: 3,33x1/3=1.11. On average, they will kill a single Space Marine.
And if 10 Marines shoot at 10 guards? 20 dice will bring us to…. 8.88 wounds (taking into account the fact that bolters ignore the Guards’ Flak Jacket). We’ll round it up to 9. Pretty impressive difference, huh?

6) Adding modifiers to the sauce

+1 to damage rolls on vehicle will mean that: 4 are now 5, 5 are now 6 and 6 well remains 6. Just keep in mind you need to base your calculations on the DIE result, then apply it to the chart.

7) 2D6 rolls

These are trickier. “A character with Ld 10 can fail on a 11 or 12 on the dice, that’s 2 chances out of 12, or 1 on 6!”. Again. No. 11 can be achieved by rolling 5 and 6 OR 6 and 5, 12 by rolling double 6. You have, by rolling 2 dice and adding them together (which is NOT the same as rolling 2 dice and looking at the results independently), 36 combinations possible. Thus a Ld10 character, rolling on his unmodified Ld, will have 3/36 chances (1 chance out of twelve after simplification of the fraction) to fail his test. 8.33%. For quick references, I made those calculations:
Final Ld / Chances to pass

2 2.77%
3 8.33%
4 16.66%
5 27.77%
6 41.66%
7 58.33%
8 72.22%
9 83.33%
10 91.66%

That means: YES there is a HUGE difference between Ld 7 and Ld 9, and NO, those critters out of synapse will not follow your masterminded plan that often. On a side note : the average roll on 2D6 is 7, and on 3D6 it's between 10 and 11.

Well, I think that sums up the question of mathammering without complicating too much. Just keep in mind : even if a single Guard has 2.7% chance to kill a Terminator, one lucky roll and your unit can be obliterated. Even if the odds are on your side, get a plan B. Just in case ;)

Dariokan

Some really basic stuff here but it's a good read for everyone to ensure we are all on the same page with our math! I've often seen people guilty of 3 which makes me cringe inside but Dariokan's final point is really important. No matter the numbers they are just numbers and whilst they give you an idea of what a unit can or cannot do, there are a lot more variables at stake and relying on averages will often see you lose.