## Getting Better at Warhammer 40,000: Understanding Probability

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For many 40k gamers, painting minis and buying books is only the beginning. At some point, they’re bound to start thinking about how they can start improving their play to win more of their games. While there’s no substitute for practice, there are a number of things players can learn to improve their play and army-building skills, which we’ll be covering in our series of articles. In this article, I’ll go over the basics of probability, and how understanding basic probability and statistics can help better predict outcomes and evaluate units in 40k, leading to smarter tactical decisions and better outcomes. And while I know that for some, the thought of digging into a bunch of math brings back unpleasant memories, I’ll try to distill this down to just the most important aspects and make it clear how you can apply all of this to your games (and the rest of your life, really!)

### Basic Statistics

One of my favorite parts of Warhammer 40k is how it allows you to finally put to use some of those parts of your grade school education you thought you’d never use “in the real world.” In this case, math. Well, more accurately statistics and probability. Properly assessing units, wargear, strategies, and board states is key to improving your game in 40k and doing so well often requires a basic understanding of statistics, or at least the common probabilities involved in the game.

Before we can properly dig in to probabilities in Warhammer 40k, we need to go over a few common statistics terms that we’ll be using and are important in statistics.

• The Probability of an event is the measure of the likelihood that the event will occur. Most of the time we talk about dice math, the probability of an outcome is equal to the number of ways it can happen divided by the number of possible outcomes. So the probability of rolling a 6 on a D6 is 1 (you can only roll a 6 one way) divided by 6 (there are 6 possible results), so the probability is ⅙, or about 16.7%.
• The Complement Rule for probability states that the likelihood of an event happening is the same as the opposite event not happening, i.e. the chance of getting a hit is the same as the chance of NOT missing. In math terms, this means the probability of an event is also equal to 1 – the probability of it not happening. So going back to our last example, the probability of rolling a 6 on a D6 is the same as the probability of not rolling a 1 to 5 on a D6. This seems trivial, but we’ll use this a lot in subtle ways.
• Independent Events in statistics are events where the probability of one occurring in no way affects the probability of the other occurring. When events are independent, the chances of both happening is just the product of the likelihood of each. So any given die roll doesn’t affect the next die roll (the die doesn’t know what just happened), so the chances of rolling two fours in a row is equal to (⅙) * (⅙), or (1/36). Note that this is not the same as the chances of rolling an 8 on 2D6, which can happen several other ways.*
*NOTE: All dice rolls are independent events! Your chances of rolling a result on a die are the same every time, no matter what you just rolled or how many 1s or 6s you rolled previously. You are never “due” for a break — this is called “The Gambler’s Fallacy”
• The Expected Value for some process is basically what we’d expect the average outcome of that process to be in the long term. You can get the expected value for a process by summing all the possible values and multiplying each by the likelihood that it will occur. So the expected value of a roll of a D6 is (1 * ⅙) + (2 * ⅙) + (3 * ⅙) + (4*⅙) + (5 * ⅙) + (6 * ⅙) = (1 + 2 + 3 + 4 + 5 + 6)/6 = 3.5. The expected value of a roll of a D3 is 2, and the expected value of a roll of 2D3 is 4 (just add the two together).
• Variance is a measure of how far results of an event can vary from their average or expected value. For our purposes, we’ll use variance to describe how broad or spread out the possible outcomes of an event are. So going back to our expected value example above, while the expected value of a single roll of a D6 is 3.5, results can go from 1 to 6, meaning that on an individual roll, we can get all kinds of outcomes. Because of this, it can often be preferable to have a flat value of 3 for damage over a D6, even though the expected value is lower – being able to predict the result can make planning much easier. A lot of the work we’ll do building armies and using stratagems will be with the goal of reducing variance.

We’ll use these concepts over and over in 40k as we use them to evaluate units and wargear and to determine the likelihood of success for some actions. One of the most common ways we’ll use statistics to do this is to calculate the damage output of a weapon based on its likelihood of wounding a target. The way you’ll see this expressed most often is as the weapon’s ability to hurt MEqs, or Marine-Equivalent targets, i.e. Toughness 4, 3+ armor save targets.

Pictured: Me rolling anything important in a game against Campbell

### Dice Math

Most of the math and statistics we’ll apply to Warhammer 40k will revolve around dice, since that’s the primary font of randomness in the game. So here’s pretty much everything you need to know about calculating probabilities around dice in 40k:

Calculating a Weapon’s Output

Generally, we can calculate a weapon’s Wound output by calculating the chance that the weapon will hit, then score a wound, then go unsaved. Taken together, this looks like:

Wound probability = (hit probability) * (wound probability) * (1 – save probability)

Then, because we may have multiple shots and do multiple points of damage per shot, we want to add those in as well.

Wound output   = shots * (hit probability) * (wound probability) * (1 – save probability) * damage

If we have a weapon with variable shots or damage, we can use expected value to substitute for those values, so we can use 2 for D3 and 3.5 for D6.

While this isn’t perfect — it doesn’t take into account factors like range for example, but it does let us do easy comparisons for weapons when we control for other factors. If say, we wanted to figure out how much damage a rapid-firing group of intercessors might do to a knight, we can do that:

5 intercessors = 10 total shots, S4 AP-1 D1

With a 3+ BS, we have a ⅔ chance to hit – there are 4 results on a D6 that will give us a hit result (3, 4, 5, and 6), which would be 4/6 and that reduces to ⅔. And at S4 vs. T8, we’ve got a ⅙ chance to Wound. At AP-1, the knight’s save is 4+, so there are 3 results that will give him a save, giving us a 3/6 or ½ chance to deal a wound.

Ok so with all of that, our math becomes:

Wound output = 10 * (⅔) * (⅙) * (½) = 10 * (1/18) = 0.56

So your intercessors rapid-firing at the knight will, on average, do about half a wound. While you might not have needed this math to tell you that shooting bolters at a knight was a bad idea, what you can use it for is to figure out how much damage you *will* need to put out to take out a specific target. On the flip-side, each lascannon you fire at the same knight has an expected value of about one wound, indicating that you’ll need to fire a lot of them to take one down. It should also serve as a sobering look at just how much you’d have to devote to drop one of the bastards.

Changing the Odds

As we start to look at abilities that improve or decrease our chances of success, we need to think about how probabilities work. Generally speaking, if we’re combining events with the word “and”, like the chances of getting a hit and a wound, we multiply the likelihood of each event occurring. If we’re talking about combining events with “or,” like when we’re talking about re-rolls (the odds of getting a hit on the first roll or the second), we add the likelihood of each event occurring.

Re-Roll Math

This gets a little bit trickier if you’re doing Re-roll math. In those cases, you have to figure out the chance of failure. Generally, the formula looks like:

Probability of an event with re-rolling failure = (chance of success) + (chance of failure) * (chance of success on re-roll)

So if you had a 3+ to hit and you could re-roll misses, your chances of hitting are:

(⅔) + (⅓)*(⅔) = (⅔) + (2/9) = (6/9) + (2/9) = 8/9, or about 89%.

If you’re one step ahead of the game, you might have noticed that getting +1 to hit with BS 3+ changes your odds to (⅚), or about 83%. As it turns out, for BS 3+, 4+, or 5+, re-rolls are better than getting a +1. Likewise, because re-rolling 1s improves your likelihood by less than (⅙), getting a +1 is better than re-rolling 1s for BS 3+, 4+, and 5+.

The exceptions here are for BS 2+, where re-rolling 1s is better than getting a +1 (because 1s always fail and most of the time you won’t have a -1 to hit), and for BS 6+, where +1 is better than a re-roll because re-rolls increase your chances of success by less than (⅙).

Charging

The other universal process in 40k that relies on variance is declaring charges. And because there are a number of abilities that allow units to arrive on the battlefield anywhere that is more than 9” away from an enemy unit, we’re particularly interested in the likelihood of successfully charging an enemy unit when they’re 9” away or more.

The first thing you have to know about charge math is that, unlike with a single D6, the outcomes of 2D6 are not equally distributed. That is, there are more ways to get 7 than there are to get 12. As a result, you’re significantly more likely to roll 7 on 2D6 than 6 or 8, and even less likely to roll 9+ or under 5. This means that the chances of rolling 9+ on 2D6 (or hitting your required charge distance after teleporting in) is about 28%.

In order to visualize this, and to put more charts into an otherwise text-heavy post on statistics, I’ve put together this handy chart showing charge probabilities by distance.

God, fuck that smug Tau so much

Going back to that 9” charge distance,  let’s take a look at how different modifiers affect your chances of pulling off a charge of 9” or more:

 Modifier Chance of Success No modifiers 28% CP Reroll (1 die) 52% Full reroll on fail (all dice) 48% Full reroll on fail or CP Reroll 58% Roll 3D6 74%

#### Applying This in Games

So now that we’ve gone through all this math, the question becomes: How do you use this? The good news is, you don’t have to sit down and write out a bunch of math to immediately apply these concepts. Here are a few ways you can think about statistics in 40k that will improve your planning and in-game outcomes:

• Use the Expected Value of an Action to determine if it’s worth doing
Once you’ve got Expected Value down conceptually, you can start thinking in terms of the outputs you can expect from a given action you’ll take. What I mean by this is, if I’ve got 10 marines about to shoot at something and my options are between Meganobz and boyz, I can do some quick thinking to figure out how valuable it’ll be to shoot at each target. In this case say my marines are more than 12″ away and moved. I can generally assume that of my 10 shots, about 6-7 will hit. Orks are Toughness 4, so of my 6-7 half, half will wound, so 3-4. Then the Ork boyz get a 6+ save so I can expect most or all of those to be kills, while I should expect to do 0 wounds to the Meganobz, maybe 1 if my opponent rolls poorly. This is an extreme example, but it’s how I might quickly think about which targets to choose based on the Expected Value of a volley of shooting.
• Know what Charge distances are actually “make-able”
We already talked about charge distances above, the single most important area of statistical application in 40k. Go back to that table. See how hard it is to make a 9″ charge without re-rolls? That’s no accident. Even with re-rolls, your chances of making that charge are a bit better than a coinflip, so if your entire gameplan rests on making a single 9″ charge, well, you should expect to lose a lot of games. Understanding this should also help you understand that you need to over-commit to reducing charge distance. The better you understand the risks you’re taking and what the long-shots are, the better you’ll be able to understand why relying on those is a bad idea. Having a re-roll for your charge distance might seem like it makes your charge a sure thing, but it still leaves you missing more than half the time.
• Reduce Variance wherever/whenever you can
A big part of your list-building and in-game strategy should be reducing variance. Re-rolls and modifiers are all about increasing the likelihood of your outcome and reducing the boom-or-bust nature of some actions. While you can’t completely remove variance, you want to reduce it as much as possible, so you’ll know what outcomes to expect and how to plan around them if they don’t go to plan. A lot of bad plans depend on high-risk, high-reward outcomes. As a corollary, avoid things that increase your variance, such as negative modifiers to hit, unless their upside is substantial enough to outweigh the risk.
• Save your CP re-rolls for high-upside situations
There are a lot of outcomes in 40k that aren’t that big a deal. You only get one re-roll per phase with your Command Point re-rolls and while it will be tempting to use them all the time, you really want to save them for big opportunities. Think multi-meltas and lascannons, not bolters. Both in terms of re-rolling hits/wounds/damage and in terms of rolling saves. yeah failing a 2+ save on an AP 0 gun sucks. But unless taking that wound will cost you big time, save your re-rolls for bigger stuff. Another way to think about this is to consider the upside to your re-roll. Will a successful re-roll meaningfully improve your chances of winning the game, or significantly reduce your chances of losing? If not, don’t waste your time.
• Use your CP re-rolls high probability events, not longshots
You’re going to have many opportunities to re-roll dice over the course of a game. When you’re picking where you want to do this, pick the moments where your re-roll has a higher chance to succeed. Don’t waste CP re-rolling things that give you a 5+ or worse unless it will literally decide the game (see ‘high-upside situations,’ above). A corollary to this is if you’re rolling for something like number of shots, you want to re-roll if you get a result of 1 or 2, and sometimes 3, but not if you get a 4 or more as your chances of getting a higher result diminish with a higher original result.
• Generally speaking, Re-rolling Shots > Re-rolling Wounds > Re-rolling Damage
More shots means more wounds means more damage. If you’re choosing where to do your re-rolls, getting more shots will create more wounds which will generate more damage, so the earlier in the chain you generate additional value, the more value you’ll generate further down the chain. There are some exceptions — you want to re-roll higher-variance dice generally, but if you’re shooting something like a battle cannon, re-rolling dice on the number of shots is better than re-rolling to wound or on the D3 damage rolls.

#### Go Forth, and Crush Your Enemies

You’ve now got all the tools you need to properly evaluate wargear, gauge strategies, and predict outcomes in 40k. If you don’t get all this in your first go, don’t worry so much – mastering this stuff takes a long time and while we’ve provided some guidelines, none of these are ironclad “rules” to follow. In-game, there will be lots of nuance and mitigating circumstances, but if you approach the game with this mindset, you’ll be better prepared to understand which circumstances matter, which don’t, and how you should respond.

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