Let's Do the Math - Dice Calculation on 2d6 and 3d6

One of the most remarkable things about the Warmachine and Hordes ruleset is its reliance on a 2d6 mechanic for attack and damage. It is hard to find another system that uses such a thing. Infinity relies on multiple d20s, 40k and Age of Sigmar use single d6 rolls (in their hundreds), Malifaux uses cards, Judgement uses custom dice pools, etc.

The 2d6 system is extremely interesting from a mathematical perspective because the resulting rolls from those two six sided dice make up something called a normal distribution, which is a statistically important construct that roughly means given all possible outcomes on the 2d6, over many many rolls, there will be a large number of 6, 7, and 8 results compared to the outlying 2 and 12 results. If forms a nice curve that looks something like this:

I am using the website www.anydice.com for these graphs - go check it out, it’s a fascinating little web tool.

As we can see, the chances of rolling exactly a 7 are the highest, with just under 17%, whereas our chances of rolling a 12 or a 2 are extremely low, at around 3% each.

The boosting mechanic in Warmachine is another fascinating piece of mathematical wizardry because adding the third dice to the probability makes the center of the curve have outcomes that are even more likely as we can see in this graph:

Now, knowing the probability for exactly a number on 2d6 or 3d6 is less important in this game than it is to know how likely you are to roll at least that number, and that graph looks like this:

Or for those of you who just want to see the numbers, here is both 2d6 and 3d6:

Alrighty, now you can see where the chances of getting at least a certain number land on these curves, and let’s talk about how to apply this math on the fly.

Let us suppose you have a Crusader that needs to kill an Ironclad. Crusaders are MAT 6 and have an initial attack at PS 18 and another initial at PS 14. For this demonstration let us say he has three focus for buying attacks and does not need to charge.

Ironclads are DEF 12 ARM 18 with 30 health, and for this purpose we will assume that we do not cripple movement until last so that their DEF stays constant at 12.

In order to calculate our odds of one rounding the Ironclad with our Crusader buddy, we can do some quick head math. The way this works is basically this - take the chance to hit, multiply it by 7 since that’s an average roll and we don’t have a “target” for damage, and subtract the difference between ARM and PS.

For this example, we have MAT 6 vs. DEF 12, so the Crusader needs a 6 to hit, or a 72% chance. If it hits with the mace, it is PS 18 vs. ARM 18, so it is at straight dice and we would multiply 0.72 by 7 which gives us 5 ish expected damage for that attack. We do that for all of the attacks, add up the expected damage, and that tells us what our output is going to likely be.

We have four mace attacks, and we have calculated the output at about 5 damage per hit so 4 times 5 is 20. Follow that up with a MAT 6 PS 14, or 0.72 *( 7 - 4) (since we are at dice -4) and that equals 2 damage. The Crusader’s expected damage output is then about 22 damage against the Ironclad.

That’s not great odds at ALL, and for those of you Android users who have the Oddsmachine app, it puts that at 19.17% for the Crusader to kill the Ironclad.

Now, what happens if we add the Choir of Menoth’s Hymn of Battle to that? We can essentially just add 2 damage to each attack, so over 5 attacks that’s 10 extra damage for an expected damage output of 32, or just barely killing it.

Now a quick note on what this means - an expected damage output that barely kills a model gives you about a 50% chance to kill that model, it doesn’t guarantee that you kill that model. I see far, far too many players run the math, go “yeah that’s my output, that thing should totally die”, and then get really upset when it does not work out because they miss one critical attack. Let me repeat - if your expected output just barely would kill a model, you are in the coin flip range of that model being dead. The odds of killing the Ironclad with the Battled Crusader according to Oddsmachine are just over 51%.

Let’s flip this around and see what a higher chance to hit does for our odds.

Say the Crusader gets +2 MAT instead of +2 Damage.

It now needs anything but snake eyes to hit, or .97 * 7 = 6.8 or basically 7 for the mace and .97 * (7 -4) = 2.9 or basically 3 for the fist. Add all this up, you get 4*7 + 3 = 31. Interesting huh? Adding +2 to our hit roll in this case is almost identical to adding +2 to the damage roll (slightly worse, but close), and Oddsmachine confirms this is just over 45% chance to work out or essentially a coin flip.

You can apply this to all kinds of things now, from odds of one rounding a heavy to killing an enemy Warcaster or Warlock (though this is more complicated thanks to transfers/overboost). If you can download Oddsmachine, I really recommend it as a good backup to your head math. Pretty soon, you start to instinctively feel how likely things are to work out based on previous experience, but be sure to run the math in your head if it’s at all important.

Before we end this little piece on dice math, let us quickly touch on boosting vs. buying. Our rough guide is if you are at Dice -4 or worse, it is better to boost attacks than to buy. If you need an attack to hit and you are needing more than a four to hit, I tend to boost. If it’s a critical, game altering, do this or lose kind of dice roll, I will boost to hit a four since going from 91% on two dice to over 99% on three dice means that I’ll win 10% more games that way.

If you need one attack to land on a model that auto kills to, for example, clear a zone or flag or charge lane, then you need to do some math and it is real dependent on the amount of boosting and buying you have access to.

Say you need an 8 to hit a model and you have an initial attack and two boosts or buys and it auto dies. Straight rolling 3 attacks gives you 3 chances at a 42% hit, which is pretty darn good odds. Boosting one hit and buying one unboosted hit gives you an 83% chance to hit and a 42% chance to hit. On paper they are roughly equivalent, but the boost is almost always going to be better.

If you only have access to one boost or buy, things get a little hazy, but needing an 8 I will usually boost.

Good rule of thumb - if it’s a 6 or lower and you have the option of 2 attacks or 1 boosted attack, go for the 2 attacks. If it’s higher than a 7, go for the boost. If it’s exactly a 7….pick one, it will vary from person to person.

That’s all we have for today! I hope this was useful and interesting.

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