You’ll often hear experienced players talking about “Mathhammer”. Some times they’ve mathed out a serious return on investment. Others how they mathhammered their force into competative shape. Still others act like mathhammer is the solution to Warhammer, both in 40K and Age of Sigmar. But if you’re just coming into the game, you’ll be asking, “What is Mathhammer?”.
Mathhammer, in short, is the use of statistics to evaluate the effectiveness of units in combat. Most of the statistical manipulation boils down to a simple expectation. How many unsaved wounds can you expect a squad to inflict each turn? From there you can derive other comparisons across units.
Mathhammer – Probabilities
Let’s start with simple expectations. How many unsaved wounds will my squad inflict. We’ll use a Space Marine Tactical Squad armed with boltguns shooting at some Death Guard Cultists as an example.
To score an unsaved wound you’ll need to have the following probabilities; hit probability (Hp), wound probability (Wp), and failed save probability (FSp). Since Warhammer strictly uses a six-sided die for rolls, all probabilities will be expressed as fractions.
Hp- hits are scored on 3, 4, 5, or 6. Your hit probability is 4/6 and simplifies to 2/3.
Wp – with S4 vs T3, wounds occur on 3 ,4 ,5 ,or 6. Again, 4/6 simplifies to 2/3.
FSp – the cultists only save on 6, so failures are 1, 2, 3, 4, and 5. They fail 5/6.
For the probability of an unsaved wound you multiply all the probabilities together.
Hp * Wp * FSp = 2/3 * 2/3 * 5/6 = 20/54 which simplifies to 10/27 or .37.
Mathhammer – Expectation
To find the expected number of unsaved Wounds from a round of fire, multiply the number of shots by the probability of an unsaved Wound you just found above.
Now, the ten man tactical squad opens fire with their Rapid Fire 1 boltguns. At 12″ they’ll fire 20 shots, at 24″ you’ll fire 10. Multiply the number of shots by the probability we calculated above.
Long range fire should generate 10 x 10/27 = 100/27 or roughly 3.7 Wounds.
Short range, on the other hand should reap 20 x 10/27 = 200/27 or roughly 7.4 Wounds.
Turning the mathhammer around for the Death Guard Cultists doesn’t look nearly as good.
Cultists hit on 4+ (Hp = 1/2), wound T4 Marines on 5+ ( Wp = 1/3), and have they need to fail their 3+ save by rolling 1 or 2 (FSp = 1/3). The probability of an unsaved wound is 1/18. At least their autoguns engage the Marines at the same ranges.
Therefore at long range you’d expect 10 x 1/18 = 10/18 = 5/9 or roughly .56 Wounds.
At short range you’re odds are slightly better as you get 20 shots. 20 x 1/18 = 20/18 or 10/9, roughly 1.1 Wounds unsaved.
Mathammer – Problems
If you’ve followed along so far, you’ll already have spotted two problems with Mathhammer. First, you never end the combat phase with unsaved fractional wounds. Second, even after a few games, you’ve seen results much higher or lower than the numbers quoted here. Let’s deal with both of them now.
Warhammer is a game of whole numbers. Regardless of the type of attack, wounds, or saves, you’ll never deal partial wounds. When you see something like the Cultists scoring 0.56 unsaved wounds, you’d read that as the whole numbers on either side of the fraction. In this case either none or 1 Wound.
You can also read the decimal as the chance of scoring an additional wound. For the cultists, it is a 56% chance of scoring 1 Wound with 10 shots. Likewise, Space Marines shooting at close range expect to score 7 Wounds with a 40% chance of an 8th.
Now, with the second problem, tabletop results differ, often widely, because they dice are random. Due to the Law of Large Numbers, actual rolled results only approach the expectation when a large number of independent trials are conducted, a very large number of trials. More rolls than you’d make in even the longest string of tournament games. I can already hear you asking why we’re doing this!
Using simple expectations gives you an idea of how effective your attacks will be against a variety of targets. While in 40K you’re able to fire a unit and evaluate results before firing again, knowing the expected unsaved Wounds lets you prioritize tough targets and knowing when attacks would fail.
Mathhammer – Expanding the Scope
Now that you understand the basis behind simple expectations you can expand the scope of your comparisons. Here’s the following stats you’ll need.
| Dice Score | Hp | Wp | FSp |
| 1+ | 6/6 or 1 | 6/6 or 1 | 0/6 or 0 |
| 2+ | 5/6 or 0.83 | 5/6 or 0.83 | 1/6 or 0.17 |
| 3+ | 4/6 or 0.67 | 4/6 or 0.67 | 2/6 or 0.33 |
| 4+ | 3/6 or 0.50 | 3/6 or 0.50 | 3/6 or 0.50 |
| 5+ | 2/6 or 0.33 | 2/6 or 0.33 | 4/6 or 0.67 |
| 6+ | 1/6 or 0.17 | 1/6 or 0.17 | 5/6 or 0.83 |
| 7+ | 0/6 or 0 | 0/6 or 0 | 6/6 or 1 |
You can construct your expectations for squads with multiple different weapons by computing the expectation for each one and adding them together.
Simple plus and minus modifiers just move the target number around. Use the new target number in your calculation.
Weapons with a random number of attacks have their own expectations
The result of a D3 is 2, while the result of each D6 rolled is 3.5.
Tactical Squad versus Death Guard Cultists again, this time with a heavy bolter and flamer. We’ll shoot at close range.
8 Boltguns (16 shots) = 16 * 10/27 = 5.9 unsaved Wounds
The probability of an unsaved Wound with a flamer follows the same formula we used previously. In this case as flamers always hit;|
Hp * Wp* FSp = 1*4/6*5/6 = 20/36 = 5/9 or 0.56
1 Flamer (3.5 shots) = 3.5 * 5/9 = 1.9 unsaved Wounds
Meanwhile the heavy bolter presents a bit of a conundrum since it has a Damage stat of 2. When firing at 1 Wound models we’d ignore the damage stat becasue spillover damage doesn’t go to the next model. If we were shooting at marines, we’d compute unsaved Wounds as Hp*Wp*FSp*Dam since those extra Damage points would effect surviving models! Also, the AP of -1 means those cultists need to roll 7+ to save.
So, against those cultists Hp*Wp*FSp = 4/6*4/6*1 = 16/36 = 4/9 = 0.44
1 Heavy Bolter (3 shots) = 3 * 4/9 = 1.3 Wounds
Just add the various expectations together to get the total number of wounds.
unsaved Wounds = 5.9 + 1.9 + 1.3 = 9.1 or 9 dead cultists
Which is a bit better than the 7.4 for 10 boltgun Marines.
Flipping it around for the Cultists again, we’ll assume it is actually a full 20 man squad with a pair of flamers! At close range we get;
18 autoguns (36 shots) = 36 * 1/18 = 2 unsaved Wounds
For the flamer Hp*Wp*FSp = 1 * 3/6 * 2/6 = 6/36 = 1/6 = 0.17
2 Flamers (3.5 shots each) = 7 * 1/6 = 1.2 unsaved Wounds
Together, the marines are looking at taking 3.2 Wounds. Effectively losing a single Marine and having another with a single Wound on them.
In this intense, short range firefight nearly half the cultists fall compared to a single Marine. The Emperor really does protect!
I hope this helps with your understanding of the math behind the games we all love to play. Until next time, here’s some extra credit reading regarding Expected Values and Probabilities.
Expected Value
Expectation and Variance
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