

Mathhammering is the act of using math to figure out how to most effectively build your list and play games of 40k.
In this article I will explain three ways to use mathhammer, firstly you can figure out a unit's average output against a certain type of target, you can figure out the average amount of input needed to destroy a unit with a certain type of weapon, finally you can figure out what the chance of success is of a given move.
Probability can be presented either through fractions or percentages, when we roll a D6 there are 6 outcomes and each dice must land on one of those 6 outcomes, 1/6. If we want to know the chance of a dice landing on one of several facings we simply count the number of facings they can land on, for a 2+ our dice can land on 2, 3, 4, 5 or 6, so 5 outcomes or 5/6. If we want to know the chance of something happening after something else happens we can multiply the two fractions together, the chance of rolling both a 6+ and a 2+ is 1/6*5/6=5/36. We can turn that into a percentage value by multiplying by 100. 5/36*100=13,88888...% We'll shorten it to 1 decimal because that's all we really need 13,9%.
To figure out the effect of a reroll on a failed roll you can multiply your number of hits/wounds by 6+the number of facings you will be able to reroll and divide by 6.
The average roll of a D3 is (1+2+3)/3=2 or a D6 is (1+2+3+4+5+6)/6=3,5.
To figure out the effect of a modifier you can multiply your number of hits/wounds by the number of successful facings after the modifier and divide by the number of successful facings after the modifier. Let's say your opponent has a 2+ Sv (failing the save on a 1) you have an AP of 1 (so the save fails on 1s and 2s so 2). You would multiply the number of unsaved wounds by 2/1. Or if you are BS 4+ and your opponent is 2 to hit you multiply the number of hits by 1/3. Modifiers don't interact with rerolls, they are simply a modifier added after the fact, it makes the effect of modifiers really simple. 2 to hit will make a BS 2+ unit 3/5 times as likely to hit and a BS 4+ unit 1/3 times as likely to hit whether they benefit from rerolls or not.
When figuring out whether a unit is worth including in your list you might want to know how effective it as at killing a certain type of unit, let's say Orks keep beating your Necrons in close combat and they take your shooting units captive so you cannot simply shoot them.
Theoryhammer: Flayed Ones should be effective against Orks because they have a large amount of attacks (3) with a decent Strength (4) and the ability to reroll failed wound rolls, they do not have any AP, but Orks only get a 6+ Sv so that is fine. Flayed Ones also have a 4+ Sv unlike Scarab Swarms so they should be fairly resilient.
First, we want to know the output of a unit of Flayed Ones, let's say we take 5 Flayed Ones from the Novokh Dynasty so they can reroll failed Hit rolls in the first round of combat.
 We take the number of models in our unit (5) multiply it by the number of attacks each model gets (3) to find the number of attacks. 5*3=15
 Divide it by 6 and multiply it by the amount of facings on a D6 that will result in a hit for our Hit rolls (WS 3s, 4s, 5s and 6s hit so 4), multiply by 6+ the amount of facings we can reroll (1s and 2s fail so 2) and divide by 6 to find the number of hits. 15/6*4*(6+2)/6=13,333333333333333333333333333333
 Divide it by 6 and multiply it by the amount of facings on a D6 that will result in a wound for our Wound rolls (S4 T4 4s, 5s and 6s so 3) and multiply by 6+ the amount of dice facings we can reroll (reroll 1s, 2s and 3s so 3) and divide by 6 to find the number of wounds. 13,333333333333333333333333333333/6*3*(6+3)/6=10
 Divide by 6 and multiply by the number of rolls our opponent can make that will result in a failed save (1s, 2s, 3s, 4s and 5s so 5) to find the number of unsaved wounds. 10/6*5=8,3333333333333333333333333333333
So with 5 Flayed Ones (17 pts per model) charging into a unit of Ork Boyz we can expect to kill 8,3 Ork Boyz (6 pts per model). Now we might want to know the return on our investment we can do that by dividing the amount of points killed by the amount of points invested (6*8,3333333333333333333333333333333)/(17*5)=0,58823529411764705882352941176471 or a 58% return on our investment, pretty good.
There are some terms to pay attention to when discussing mathhammer we talk about a number of different profiles you might want to be able to kill with a unit, GEQ W1 T3 Sv5+, MEQ is W1 T4 Sv3+, Terminator is W2 T4 Sv2+ 5++ (5+ invulnerable save), VEQ is infinite wounds T7 Sv 3+, KEQ is infinite wounds T8 Sv 3+ 5++. We treat Vehicles and Knights as effectively having infinite wounds to make the math easier, in reality if you shoot lascannons (D6 damage) 1 at a time into a Knight (24 wounds) until you kill it you are likely to do some amount of overkill with the last shot, you can avoid this by firing your multidamage weapons first and using your lower damage weapons to finish a multiwound model off.
Next we can figure out how much of a beating our Flayed Ones can take, to figure out how much of a beating our unit can take we'll work backwards.
 We take the number of unsaved wounds the unit is allowed to take (5).
 Multiply it by 6 and divide by the number of facings on a D6 that will result in a failed save (1s, 2s and 3s so 3). 5*6/3=10
 Multiply it by 6 and divide by the number of facings on a D6 that will result in a wound (S4 T4 4s, 5s and 6s so 3). 10*6/3=20
 Multiply it by 6 and divide by the amount of facings on a D6 that will result in a hit (WS 3+ 3s, 4s, 5s and 6s so 4). 20*6/4=30
 A unit of Orks will need to make 30 attacks on average to wipe out our Flayed Ones. Orks can have between 3 and 5 attacks.
 Assuming they have 3 attacks the Orks will need 10 Ork Boyz or 60 pts worth of models, that will yield our enemy a 141% return on their investment. Much better than we expected for our Flayed Ones.
 Assuming they have 5 attacks the Orks will get a 236% return on their investment. This just puts the Flayed Ones to shame, I guess we can go back to the drawing board, a counteroffensive with Flayed Ones is probably not a good tactic for fighting Orks. We could contrast and compare with Scarab Swarms to figure out if one is better than the other at this role.
The last thing we can compute with mathhammer is the chance of an action succeeding, like the likelihood of rolling 7+ on two dice, 1/6 of the time you roll a 1 and need a 6 to roll 7+ (1/6*1/6) 1/6 of the time you roll a 2 and need a 5 or 6 (1/6*2/6) to succeede and so on if we roll a 6 any roll on the second dice will result in a roll of 7+.
 1/6*1/6+1/6*2/6+1/6*3/6+1/6*4/6+1/6*5/6+1/6*6/6=58%
We can also calcuate the chance of rolling two 6s 1*1/6/6, the chance of rolling either two 1s or two 6s is 1*1/6/6*2.
Or the likelihood that a Predator with 4 lascannons (BS 3+ S9 AP3 D6 damage) destroying a Venom (1 to hit T5 Sv4+ 5++). First we figure out the likelihood of nothing hitting 3*3*3*3/6/6/6/6=6,25% because there is a 3/6 chance of failing to hit and the last dice we roll has to be fail. The chance of everything hitting is 3*3*3*3/6/6/6/6=6,25% because the last dice we roll has to be a failure after rolling three failed hits. The chance of hitting once is 3*3*3*3/6/6/6/6*4, we multiply by four because it can be any of our four dice that results in that one successful hit. Likewise with three hits, because any one of the four dice has to land on a 13, but all the others have to be hits. 3*3/6/6 of the time our first two shots will be hits and the remaining two will fail 3*3/6/6 of the time and we will have a total of two hits, likewise with hitting twice and then having to fail twice, if we subtract the chance of hitting twice or failing twice we get the chance of hitting exactly once with two shots 13*3/6/6*2 the chance of that happening twice in a row is (13*3/6/6*2)^2.
 Chance of hitting 0 times 3*3*3*3/6/6/6/6=6,25%
 Chance of hitting 1 time 3*3*3*3/6/6/6/6*4=25%
 Chance of hitting 2 times 3*3/6/6*3*3/6/6+3*3/6/6*3*3/6/6+(13*3/6/6*2)^2=37,5%
 Chance of hitting 3 times 3*3*3*3/6/6/6/6*4=25%
 Chance of hitting 4 times 3*3*3*3/6/6/6/6=6,25%
 Chance of wounding 0 times with four dice on 3+ is 2*2*2*2/6/6/6/6=1,23%
 Chance of wounding 1 time with four dice on 3+ is 4*2*2*2/6/6/6/6*4=9,88%
 Chance of wounding 2 times with four dice on 3+ is (1(2/6)^2(4/6)^2)^2+2*(4/6)^2*(2/6)^2=29,63%
 Chance of wounding 3 times with four dice on 3+ is 4*4*4*2/6/6/6/6*4=39,51%
 Chance of wounding 4 times with four dice on 3+ is 4*4*4*4/6/6/6/6=19,75%
 0 wounds on three dice 2*2*2/6/6/6=3,70%
 1 wound on three dice 4*2*2/6/6/6*3=22,22%
 2 wounds on three dice 4*4*2/6/6/6*3=44,44%
 3 wounds on three dice 4*4*4/6/6/6=29,63%
 0 wounds on two dice 2*2/6/6=11,11%
 1 wound on two dice 1(2/6)^2(4/6)^2=44,44%
 2 wounds on two dice 4*4/6/6=44,44%
 0 wounds on one dice 2/6=33,33%
 1 wound on one dice 4/6=66,67%
 0 wounds on zero dice = 100%
 Chance of wounding 0 times with 4 attacks 0,0625*1+0,25*0,3333+0,375*0,1111+0,25*0,0370+0,0625*0,0123=19,75%
 Chance of wounding 1 time with 4 attacks 0,25*0,6667+0,375*0,4444+0,25*0,2222+0,0625*0,0988=39,51%
 Chance of wounding 2 times with 4 attacks 0,375*0,4444+0,25*0,4444+0,0625*0,2963=29,63%
 Chance of wounding 3 times with 4 attacks 0,25*0,2963+0,0625*0,3951=9,88%
 Chance of wounding 4 times with 4 attacks 0,0625*0,1975=1,23%
 Chance of 4/4 5+ saves failing 4*4*4*4/6/6/6/6=19,75%
 Chance of 3/4 5+ saves failing 4*4*4*2/6/6/6/6*4=39,51%
 Chance of 2/4 5+ saves failing 14*4*4*4/6/6/6/64*4*4*2/6/6/6/6*44*2*2*2/6/6/6/6*42*2*2*2/6/6/6/6=29,63%
 Chance of 1/4 5+ saves failing 4*2*2*2/6/6/6/6*4=9,88%
 Chance of 0/4 5+ saves failing 2*2*2*2/6/6/6/6=1,23%
 Chance of 3/3 5+ saves failing 4*4*4/6/6/6=29,63%
 Chance of 2/3 5+ saves failing 4*4*2/6/6/6*3=44,44%
 Chance of 1/3 5+ saves failing 4*2*2/6/6/6*3=22,22%
 Chance of 0/3 5+ saves failing 2*2*2/6/6/6=3,70%
 Chance of 2/2 5+ saves failing 4*4/6/6=44,44%
 Chance of 1/2 5+ saves failing 4*2/6/6*2=44,44%
 Chance of 0/2 5+ saves failing 2*2/6/6=11,11%
 Chance of 1/1 5+ saves failing 4/6=66,67%
 Chance of 0/1 5+ saves failing 2/6=33,33%
 Chance of causing 4 unsaved wounds with 4 attacks 0,0123*0,1975=0,24%
 Chance of causing 3 unsaved wounds with 4 attacks 0,0123*0,3951+0,0988*0,2963=3,41%
 Chance of causing 2 unsaved wounds with 4 attacks 0,0123*0,2963+0,0988*0,4444+0,2963*0,4444=17,92%
 Chance of causing 1 unsaved wounds with 4 attacks 0,0123*0,0988+0,0988*0,2222+0,2963*0,4444+0,3951*0,6667=41,83%
 Chance of causing 0 unsaved wounds with 4 attacks 0,0123*0,0123+0,0988*0,0370+0,2963*0,1111+0,3951*0,3333+0,1975=36,59%
 Chance of causing 6 damage with 0 unsaved wounds =0%
 Chance of causing 6 damage with 1 unsaved wound =16,67%
 Chance of causing 6 damage with 2 unsaved wounds 1/6*2/6+1/6*3/6+1/6*4/6+1/6*5/6+1/6*6/6+1/6*6/6=72,22%
 Chance of causing 6 damage with 3 unsaved wounds 1/6*36/36+1/6*36/36+1/6*36/36+1/6*35/36+1/6*33/36+1/6*31/36=95,83%
 Chance of causing 6 damage with 4 unsaved wounds 1/6*216/216+1/6*216/216+1/6*216/216+1/6*216/216+1/6*215/216+1/6*212/216=99,61%
 Chance of killing a Venom with a laslas Predator 0,0024*0,9961+0,0341*0,9583+0,1792*0,7222+0,4183*0,1667+0,3659*0=23,42%
So there is a bigger chance of causing 0 damage than causing 6 damage with four lascannons against a Venom. The average amount of damage we do with basic mathhammer 4/6*3/6*4/6*4*(1+2+3+4+5+6)/6=3,11 damage, but that number is off because we are ignoring the possibility of overkilling it, but we can also see that even if we on average only halffinish the Venom, it still has a 23,42% chance of being destroyed.
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