General Warmachine Guide: Dice Math
I've got you now... My opponent passes the turn, Butcher1 is on 0 focus and I don't think he realized how far my Aiakos Stalker can go. I measure it out and sure enough I can walk 9, jump 5 and be in melee with a 0 focus Butcher! Skarre1 can feat on him before he goes for +5 str and I'm going to win! Wait, I am going to win right? I got a jack on his caster under Skarre's Feat! That's not winning? I have to roll dice? Well surely I'll still win, my Stalker has a good Mat, I can load him up with focus and he has +5 str from the feat. Hmm now you've got me wondering, do I win if I do this? Understanding the dice math is imperative to being able to make solid decisions about sending pieces in. Is this really a Killing blow, is it a piece trade, or a piece sacrifice? Lets take a look.
Okay, lets start with the basics. The gorgeous thing about Warmachine is it's core dice mechanic. By rolling 2 dice and taking the sum of them we end up with a really nice bell curve, otherwise known as a normal distribution. With 2 dice the most common result is 7, being able to be equaled with 6 unique combinations. The further away from 7 either way we go the lower the odds of hitting that number are. In Warmachine we commonly have a target number we want to roll equal to or higher. The stacked bar graph below has the chance to roll the exact number in blue, and the chance to roll below that number in red. They are stacked so the sum of the two is the chance to roll that or below, which is generally how Warmachine works.
So my stalker is Mat 7, Butcher's Defense is 14, I need a 7 or higher to hit, from the graph I see I have a 16.67% chance to roll a 7 and a 41.67% chance to roll higher, for a combined odd of 58.34% chance to hit. Hmm, thats not as good as I thought it'd be. Well lets see what each of those swings would do for damage. I'm PS 17 thanks to Pskarre's feat he's arm 18. From the chart I see the Average roll is a 7, so each hit would grant me an average of 6 damage done to his caster.
With 2 initials and 2 focus on the Stalker I have 4 attacks, those attacks hit 58.34% of the time granting me 2.34 hits per 4 attacks. Those 2.34 hits grant me 14 points of damage. Butcher would LIVE!?! GAH MATH SUCKS! ALL IT DOES IS RUIN MY PLANS! Hold on there. Another great thing about the Warmachine is the boost mechanic. Boosting is adding a third die to the roll and summing all 3. Not surprisingly it dramatically changes the math!
Now adding another die does a few things. The number of possible combinations went from 36 to 216. It raises our lowest possible roll from a 2, to a 3 and the odds of rolling the lowest possible roll go from 2.78% to .46%. That's less than half a percent, or 1 in 216 rolls. The same is true for our ceiling going from 12 to 18. TT, you're stating the obvious again. MOAR DICE IS MOAR BETTER! BRING BACK BANES! You're right but I'm going somewhere with this. It changes the center of our bell curve from 7 to 10.5. That's huge,a 50% increase in value. Lets assume Pskarre can get close enough for Dark Guidance to benefit the Stalker on Butcher. This grants the stalker an additional die to hit. We still need that 7 to hit, but instead of 58.34% chance to do so, I'm now 90.74% chance to do so. Where previously I was expected to get just over 2.34 hits out of my 4 swings, now I expected to get 3.63 hits. This takes my projected dmg from 14 points to, just under 22. Which means Butcher Dies! YAY Maybe Math Isn't All Bad After All!
Now how on. Obviously I can't hit 3.63 times, so really this Butcher game winning run comes down to do I think I can hit with a 90% four times? GAH MOAR MATH WHY!?! Well, average dice say 3 rolls leaves him with 2 boxes left. What are the odds I hit all 4 times? Well we multiple the odds 90%*90%*90%*90% (odds^#rolls) or close to 66% of the time. So while most of the time we kill Butcher, we should be ready for this not to work as well, since thats over 30% of the time. If we stack an armor debuff as well, following Cryx Commandment #1 we can be sure he'll die as the expected dmg per hit then goes from 6 to 8 for Darkshroud for Example, and we're no longer dependent on the 4th hit. To see what our chances of hitting 3 out of 3 times we take the same equation 90%*90%*90% or 72%. But in our last situation we only need to hit 3 out of 4 times. To do this takes a little more work (promise this is the last bit of math for the day). We have the Probability of the success, 90.74% (p). We know we have 4 rolls (n). We know we need 3 (x1) or 4 (x2) to be successful. We can use binomial distribution to calculate the probability of x1 and x2 then simply add them together. TT, WTF IS A BINOMIAL DISTRIBUTION AND WHY WOULD I DO THIS IN AN EVENT? Okay you can print out some distribution charts for Quick references? Nah bro no way I'm doing this. I'm down for this bell curve but this isn't fun anymore. Okay... I went a head and calculated it though, it gives us a 95.46% chance to hit 3 or more times when the chance to hit is 90.7% to hit once, and we have 4 swings.
Okay, so maybe the end bit is more than the average amount people want to calculate on a death clock. That being said I strongly encourage players new or old to have some awareness of the probability of getting a given number on any roll. I've seen printed sheets or apps to do dice odds several times in events and think it's fantastic! Next time maybe we can talk about total odds of an assassination run, and min deviations to stay on target! MORE MATH! And for those of you that can't be bothered with the odds, I salute you.