MathHammer
Overview
MathHammer at its most basic refers to the application of statistics to judge army composition decisions or determining how many attacks of a given type you would need to apply to a unit to kill it on average or determining the likelihood of killing a unit given that you put a certain number of attacks into that unit.
As the name implies it's frequently used in Warhammer 40k, but can apply to any chance-based game. The type of math we employ in Warhammer 40K is also known as Bayesian Probability.
Causing an Unsaved Wound, the 216 method
Being built around 3 successive rolls of 1d6 (To hit, To Wound, Save), 40k lends itself to a very quick and effective method for getting a % chance of causing an unsaved wound, without having to resort to spreadsheets. The 216 method (6^3) simply totals up the chances of getting a result you want, then divides them by the total number of results that can occur, which for rolling 3d6 is 216.
This example will use a Chaos Marine attacking a Guardsman in close combat
- Total the number of roll results that result in a hit - 3+ so 4
- Total the number of roll results that result in a wound - 3+ so 4
- Total the number of roll results that result in a FAILED save - 5+ so 4
- Multiply them together: 4x4 = 16, 16x4 = 64, this is the number of possible dice rolls on 3d6 that result in what you want, i.e. a casualty.
- divide the number you got from step 4 by 216, in this case 0.29~, which is the % of causing an unsaved wound.
This gives a result of one attack having a slightly worse than a 1 in 3 chance of killing a Guardsman in close combat.
Feel No Pain or Ignoring Unsaved Wounds
Some units have abilities or can benefit from psychic powers that let them ignore unsaved wounds on a roll of 6+ or 5+. To calculate the effect of this divide 6 by the number of rolls that will go through the Feel No Pain ability, with a 6+ that would be 6/5=1.2 and with a 5+ it is 6/4=1.5. So assuming the damage of the weapon is 1 then a 6+ FNP increases durability by 20% and a 5+ by 50%.
With multiple damage it becomes a little more complicated. A given unsaved wound with a damage characteristic of x has a 1/6, 1/36, 1/216, 1/1296, 1/7776, 1/46656 of being reduced by 1, 2, 3, 4, 5, 6 damage by a 6+ FNP or 1/3, 1/9, 1/27, 1/81, 1/243, 1/729 by a 5+ FNP.
Let's say your Deathshroud Terminators with 2 wounds per model and a 5+ FNP fail 10 saves with D3 damage. The first wound will cause 3 damage 27/81 times, 2 damage 27/81 times and 1 damage 27/81 times.
- 3 damage gets reduced to 0 damage 1/27 times, 1 damage 6/27 times, 2 damage 12/27 times, 3 damage 8/27 times.
- 2 damage gets reduced to 0 damage 3/27 times, 1 damage 18/27 times, 2 damage 6/27 times.
- 1 damage gets reduced to 0 damage 9/27 times, 1 damage 18/27 times.
- 13/81 times the first wound does 0 damage, 42/81 times the first wound does 1 damage, 18/81 times the first wound does 2 damage, 8/81 times the first wound does 3 damage.
- To make up for the wounds that do no damage we multiply by (81+13)/81
- To make up for the wounds that cause 1 damage we multiply by (81+42)/81
- 10*(81+13)/81*(81+42)/81=5.67 Deathshroud Terminators die on average.
Re-rolls
An ability to re-roll will multiply the number of results that result in a hit or a wound by (6+x)/6, where x is the number of facings on a hit roll you can re-roll, so if you re-roll 1s x is 1, re-roll 1s, 2s, 3s and 4s x is 4.
Chaos Space Marines within 6" of a Chaos Lord can re-roll 1s to hit so you multiply their number of results that result in a hit by 4*(6+1)/6=4.667
The efficacy of re-rolls cannot be understated, a Chaos Lord increases the damage output of nearby units by 17%, that means that if you have 6 times the value of the Chaos Lord in nearby units then the Chaos Lord will be effective for his pts simply based on his Aura, note that as you lose units the Aura will become less and less effective and Characters are often less durable for their pts than other units, so while a Chaos Lord is amazing while you have 10 times the number of pts within range of his aura, if you only have 2 times the number of pts within range of his aura he needs to do more work himself to justify his pts cost and inclusion in your army.
Required Attacks to Kill a Unit on Average
Divide the number of unsaved wounds you want to cause by the chance of causing an unsaved wound with an attack, say you want to kill an Infantry Squad with 10 Guardsmen in melee with your Chaos Space Marine unit in melee you would divide 10 by 0.29 and get 34.48.
Average Damage Output
To calculate the average damage output of a unit simply multiply the number of attacks with a given weapon by the chance of causing an unsaved wound and the average damage of the weapon. Say you have a unit of 10 Chaos Space Marines with 21 attacks, you want to know how many Guardsmen they kill on average. 21*0.29=6.09.
Points per Wound
To calculate the effectiveness of a given unit you can divide the average damage output by the cost of the unit. 10 Chaos Space Marines cost 130 pts, so each unsaved wound in close combat costs you 130/6.09=21.35 pts.
You can go one step further and divide it by the cost of a wound on a model that you are killing. 10 Guardsmen cost 40 pts so 1 is 40/10=4 pts, 21.35/4=5.34.
Now you can compare the squad to other units available to you, 5 Khorne Berzerkers for example pay 7.91 pts per unsaved wound they cause assuming they don't lose any models when the Guardsmen strike back before they can fight a second time. That means Khorne Berzerkers with chainaxes are 2,69 times as effective at killing Infantry Squads in melee compared to a Chaos Space Marine unit with chainswords. For this reason and because attacking twice also gives a lot of tactical flexibility Chaos Space Marines with chainswords are never used, on the other hand Khorne Berzerkers get used a lot more relative to the Chaos Space Marines. You also need to factor in how vulnerable a unit is, in this case Khorne Berzerkers pay 30% more per wound, but if you are paying for a Rhino to transport and protect your melee unit, suddenly you are paying very little extra to have Khorne Berzerkers instead of Chaos Space Marines and the durability issue is mitigated by the Rhino.
Why you would want to do this
So why exactly would you want to recreate statistics 101 in your hobby time? Because GW didn't. Running Mathhammer against your army list will show up the truly junk options that were crammed in your codex, and in turn the absolutely insane power breaks that have no bearing on units points cost. It can however be very useful in avoiding trap units, and putting together viable counters if you are struggling with army composition.
Beware!
Mathhammer, like employing the power of the warp, can be a force for good when used with discipline and restraint. But reckless mathhammer is a step on the path to damnation.