LOTS - Dice Odds
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- This article is part of Warmachine University's Learn Objectives, Tactics, & Strategy (LOTS) series, which is "Advanced Training" aimed at players who understand the rules, and now want to improve their gameplay.
As of 2020.12 the LOTS articles are a WIP. If you have a question that isn't answered, or a topic you'd like to see added, we'd like to hear about it. Post your thoughts on the Talk:Learning to Play the Game (LPG) page.
In this article I'm going to try and teach you how to do maths "badly" so you can do it quickly. There are plenty of websites/apps out there that can let you work out the exact odds of something to 10 decimal places, but this article is all about being in-exact.
You still will need to do some maths, sorry. I'm trying to make it easier, but I can't delete it. If you hate maths and don't want to learn then you can stop reading now and skip to the next lesson.
Simplifying the To-Hit Roll
The chances of rolling any given number on 2d6 is shown with the red bar graph diagram. The odds of rolling at least a given number, say a 5+, is given by the green line on the second diagram. We're going to simplify that to the blue dashed line.
Before we start simplfying, let's take a closer look at the exact odds of that green line.
|Roll needed|| 2d6 |
Odds of getting it
Well that's garbage info. Sure, it's all correct and accurate, but it's garbage for what we want to do.
So first thing, we're going to lump 3+ 4+ 5+ together as "almost certain" and we'll lump 10+ 11+ 12+ as "snowball's chance in hell". Second, we'll round all the numbers.
Let's take a look at it now:
|Roll needed|| 2d6 |
Odds of getting it
|3+ to 5+||80 %|
|6+ to 7+||60 %|
|10+ to 12+||5 %|
Doesn't that look nicer? You've got a lot less to memorise in the first place, and the percentages are following a much simpler pattern.
A similar simplified chart can be made for a 3d6 roll.
|Roll needed|| 3d6 |
Odds of getting it
|4+ to 7+||90 %|
|8+ or 9+||70 %|
|10+ or 11+||50 %|
|12+ or 13+||30 %|
|14+ to 18+||5 %|
Simplifying the Damage Roll
You can simplify this two ways, depending on what your target is.
- If your target has a heap of hitpoints, like a heavy warjack, you probably want to estimate the total damage you're going to do to it.
- If your target has very few hitpoints, like a single-wound infantry model, you probably want to know the odds that you'll kill it with a single attack.
Estimate total damage
- The average roll on 2d6 is 7.
- The average roll on 3d6 is 10.5
- The average roll on 4d6 is 14
- You're gaining 3.5 per die.
Total damage is kinda easy: if you've got 3 guys doing <2d6 minus 5> damage then on average: they'll inflict <7 minus 5> equals <2 damage each>. So six damage total.
What I've just said is inaccurate because of the way that, if you fail to break ARM, you deal zero damage. But the inaccuracy only crops up when your POW is a lot lower than their ARM, such as a bunch of light infantry trying to damage a heavy Khador warjack and, frankly, if you're doing that then you should already know that your odds are pretty low. (If you want to know more about this inaccuracy, we cover it more at the bottom of this article.)
Odds of getting at least [X] damage
This one is really easy - you work out what you need to roll, then look it up in the simplified odds table, above.
Say you're POW 10, and you're attacking an ARM 15 model with 2d6 damage.
- If the target has 1 hitpoint, you need to roll a 6+. That's got a 60% chance.
- If the target has 5 hitpoints, you need to roll an 11+ to kill it in one attack. That's got a 5% chance.
- If the target has 5 hitpoints, and you have a multiple attacks, then you need to make some quick assumptions.
Do you want to assume you do 3 damage with two attacks (each roll needs an 8)+, or 2 damage with three attacks (each roll needs a 7+), or 1 damage with five attacks (each roll needs a 6+)?
Once you've made that assumption, you can look it up on the simplified odds table.
Rerolls & Similar
Putting it Together
Three Tricks that Mathematicians Hate
To put the odds together there are three maths things you need to know.
- You need to know your basic multiplication table. That is 1x1 through to 9x9.
- You need to know how to move zeros around.
- If you're working with numbers bigger than 1, you smush the zeroes together to the right.
- If you're working with numbers less than 1, you smush the zeroes to the left.
- 20 x 20 = 400
- 2 x 2 = 4
- 0.2 x 0.2 = 0.04
- You need to know how probabilty works.
- Probablity works by multiplying stuff together.
- If you flip a coin twice in a row, the odds of getting two heads is: the odds of getting heads the first time (1 in 2) multipled by the odds of getting odds the second time (1 in 2) equals (1 in 4).
- Similarily, if you roll 2d6 twice in a row (to hit and to damage), the odds of getting 7+ twice in a row is: (approximately 60%) multipled by (approximately 60%) equals (approximately 36%).
Also, you should know, probabilities are always less than 1. Whether they're written (50%, 1 in 2, ½, or 0.5) really this is just different ways of writing (0.5).
Doing the Maths
Buy or Boost
Total Damage Inaccuracy
A lot of players will tell you "The average roll on 2d6 is 7. If you've got 3 guys doing <dice minus 5> damage then they'll inflict 2 damage each, on average. So six damage total." which sounds correct but is actually wrong.
Why is it wrong? Think about it:
- If you've got 3 guys doing <dice minus 7> damage, how much damage will they do on average? Zero?
- If you've got 3 guys doing <dice minus 9> damage, how much will they do on average? Negative six damage?!
Do you see the problem now?
- The incorrect version would be true if rolls that fail to break ARM inflicted negative damage, but they can't, so it starts falling apart when POW is much lower than ARM.
To be fair, the incorrect version is accurate if your POW is close to their ARM, and only a little inaccurate at much lower POW (see the tables below).
Average = 7
Average = 10.5
|POW vs ARM||Incorrect damage||Actual damage||Incorrect damage||Actual damage|
|Dice minus 2||5||5||8.5||8.5|
|Dice minus 4||3||3.1||6.5||6.5|
|Dice minus 6||1||1.6||4.5||4.6|
|Dice minus 8||(-1) ??||0.6||2.5||2.8|
|Dice minus 10||(-3) ??||0.1||0.5||1.5|
|Dice minus 12||(-5) ??||0||(-1.5) ??||0.6|