# Understanding Probability and Using it to Improve Your Game

This article was originally published in April 2019. It has been updated as of August 2021.

For many 40k gamers, painting minis and buying books is only the beginning. At some point, you’re bound to start thinking about how you can start improving your play to win more 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, many of which we’ve covered in our Start Competing series. 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, 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.” Not that tabletop wargaming is “the real world,” but it is an application of some of those math skills you thought you’d never use. Well, not so much “math” as “statistics and probability,” if you ever took a course on those. Generally speaking, statistics and probability are a major part of any game with dice or randomness. Properly assessing units, wargear, strategies, and board states is key to improving at these games and doing this 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. As much as people would like to believe otherwise, the result of any given die roll has no effect on any other, so the chances of rolling say, 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 single 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 expected values for each D3 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 as we use them to evaluate units and equipment options 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 odds of some action succeeding (like a charge in 40k) or to evaluate the damage output of a weapon based on its likelihood of wounding a target. The way you’ll see this latter piece expressed most often in 40k is as the weapon’s ability to hurt MEqs, or Marine-Equivalent targets, i.e. Toughness 4, 3+ armor save targets, though you can make up targets for any faction or game system to use in this, as we’ll see later on.

## Dice Math

Most of the math and statistics we’ll apply to board games and tabletop wargames will revolve around dice, since that’s the primary font of randomness in those games. Here are a few basic things to know about randomness and some of the common ways it pops up in games.

### The Results of  Single Die Roll are Evenly Distributed, but the Results of Multiple Dice Are Not

Let’s start with one of the most basic concepts, which new players can often get wrong. If you’re rolling a single die, the chances of each result are the same – the odds of any result on a single six-sided die are one in 6, or 1/6 – as we mentioned before, there’s only one way to roll any given number. But if we roll 2D6, those chances immediately change. Two dice give us 11 possible results, but 36 possible ways to roll the two dice – We could roll snakeyes (1/36 odds), or a 2 on one die and a 1 on the other, or a 1 on the first die and a 2 on the other, which is a different result with the same total, giving us two ways to get a result of “3”, or 1/12 odds. There are actually six different ways to roll a 7 on 2D6, giving you 1/6 odds of rolling a 7 (16.7%), making it the most likely result on 2D6 by a significant margin. In fact, 7 is the expected value of a 2d6 roll, and you’ll find that the more dice you roll, the greater your odds of rolling the expected value or something close to it.

These concepts are also true in games where you’re rolling multiple d8s or larger dice, such as in X-Wing. In that game you have blank results that make the math easier but if you’re thinking about the odds of pushing damage through against an opponent rolling dodges, you can apply these same concepts.

In wargames, this can immediately be applied to effects that let you roll more dice – rolling more dice smooths out your results, reducing your variance and giving you a higher floor for your results.

### A Common Example: Estimating a Weapon’s Output

Generally, we can calculate a weapon’s output by finding the chance that the weapon will hit its target, score a wound, and then bypass the target’s armor and deal damage. In Warhammer 40,000, that process looks like this:

(Chance to hit) * (Chance to Wound) * (Chance for the shot will go unsaved) * (damage the shot will do or casualties it’ll cause)

This ends up being:

(Hit probability) * (wound probability) * (1- save probability) * (damage)

Let’s say we want to figure out, on average, how much damage a squad of 10 Intercessors shooting at a Drukhari Raider will do in Tactical Doctrine. Assuming no modifiers right now, Marines hit on a 3+, which is four results out of 6 on a D6 (3, 4, 5, and 6), giving them a 4/6 = 2/3 chance of hitting. They wound on a 5+, which is 2 out of 6 results, or 2/6 = 1/3 probability of wounding. And in Tactical Doctrine they’re AP-2, so the Raider will usually have a 5+ invulnerable save to take, meaning they’ll save on 2 out of 6 results (5, 6), so the chances of saving are 1/3 and the odds of a wound going unsaved are 4/6 = 2/3. And bolt rifles do 1 damage. That gives us the following calculation for one shot:

expected damage out put = (2/3) * (1/6) * (2/3) * (1) = 0.15

So on average, a single marine trying to chip away at a Raider won’t do much. Your odds of actually doing damage are pretty low on a single shot. But, take a squad of 10 popping off 20 shots, and suddenly your average damage is around 3 wounds.

### The Order of Your Rolls Generally Doesn’t Matter

Here’s a fun one that’s not so intuitive – remember when we mentioned that dice rolls are statistically independent events? Well, they are, and because of that, the way you calculate the odds of two things both occurring is just to multiply their odds together. But here’s the fun thing: Because of the commutative property of multiplication, those can be done in any order without changing the odds. So if you wanted to roll wounds and saves before hits, you could do that and it wouldn’t change your odds. That said, there can matter sometimes, but it’s usually only when modifiers are involved and in those cases it’s based on what’s being modified and by how much.

### Modifiers and You

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 a re-roll-able 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, if your target number is a 3+, 4+, or 5+, having full re-rolls is 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 when your target number is 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).

#### When to Re-roll a Result that isn’t a strict failure

When you re-roll a result you give yourself a second chance to beat your original result, but the odds of getting any particular result on the re-roll are no different from the original roll. Sometimes re-rolling is a no-brainer; you’ll have failed to achieve some target number and as a result attempting a re-roll is always sensible if the roll is important. On the other hand, there may be times when the result matters but isn’t strictly a success or failure – one example is re-rolling the number of shots for a weapon or rolling the distance to Advance. In each of these cases if you aren’t looking to achieve a specific results, the general rule is “re-roll the result if it’s lower than the expected value of the roll.” So on a D6, you’d re-roll results of a 1, 2, or 3 (and in some cases you might keep 3 if that’s sufficient for your needs and a 1 or 2 will be unacceptable), and on 2D6 you’d re-roll results less than a 7 (again, considering 6s as circumstances dictate).

There’s some easy math we can do here to look at the expected value based on our re-roll rules. Let’s say we’re rolling a D6 and choosing to re-roll results less than 4, which will occur 1/2 the time. Our new expected value is:

Expected value = (1/2) * ((4 + 5 + 6)/3) + (1/2) * (3.5) = (1/2)*(5) + (1/2)*(3.5) = 4.25

On the other hand let’s say we’re re-rolling results of a 4 as well, trying to get better results, i.e. “getting greedy”:

Expected value = (1/3)*((5+6)/2)) + (2/3)*((1+2+3+4)/4)) = (1/3)*(11/2) + (2/3)*(10/4) = 3.5

We’re no better than we started at this point, and we’ll find the same results if we opt to keep results of a 3. So when your desired outcome is just “getting generally higher results,” the rule of thumb to use is to re-roll results below your expected value.

#### +1 and -1 Modifiers

There are plenty of different modifiers and how good they are usually depends on what your starting stat/roll is and how much you’re changing it by. Getting +1 to hit is a bigger deal when it’s doubling your odds of success than when it’s marginally improving them. The impact of a modifier depends on the initial probability of the roll being modified. Unlike re-rolls, modifiers are not static. Their influence is greater when the probability of success is already low, and the magnitude of the influence is the same regardless of whether the modifier is positive or negative. The table below shows the impact of a positive and negative modifier on probability.

 Roll Initial Probability -1 Modifier Probability Delta +1 Modifier Probability Delta 2+ 83% 67% -20% 83% +0% 3+ 67% 50% -25% 83% +25% 4+ 50% 33% -33% 67% +33% 5+ 33% 17% -50% 50% +50% 6+ 17% 0% -100% 33% +100%

Remember that the bonus from rerolls is a flat 16.7% improvement, which means that modifiers are always going to have a greater impact on probability than rerolling a single number. Armor save modifiers are by far the most common modifier seen in the game, and since most models have at least a 6+ save the difference between AP 0 and AP -1 is significant. This can be seen in the chart below.

#### Things to Remember

• The impact of re-rolling a fixed number on a single six-sided die is a flat 16.7% increase in probability per number; re-rolling 1s or 2s would improve your chance of success by 33%.
• The effect of rerolls is multiplicative between rolls. If you were re-rolling 1s to hit and 1s to wound, that’d give you a cumulative improvement of 36% in your odds of pushing through damage
• The effect of a modifier is inversely proportional to the probability of the roll; the worse your chances of success, the greater the impact of a modifier to that roll will be.
• Outside of edge cases where probability cannot be changed, modifiers tend have a greater impact on probability than re-rolls

#### Driving Rolls and Sequencing

Some rolls directly influence the final outcome by determining either the number of wounds, or the number of attacks, that are dealt to the target. From a mathematical perspective these “driving rolls” have an equal influence on the total wounds dealt to the target. In other words the total damage dealt to a target will be the same if you double the number or shots or double the damage. On the other hand, when rolled in a sequence you may be able to improve those odds during the sequence, in which case you’ll be able to have an active effect on the variance by modifying rolls earlier in the string.

For example, let’s say you were attacking with a weapon that made D6 attacks, doing D6 damage, and you had the ability to re-roll either the number of attacks or the amount of damage. You won’t be able to choose to re-roll shots when you’re re-rolling damage, and those rolls come first. In both cases your expected damage is the same – let’s say it’s a game of Warhammer and you’re wounding a 3+ and the enemy has no save against the attack, your expected damage is:

expected damage = (expected shots) * (chance to wound) * (chance of an unsaved wound) * (expected damage)

That gives us:

expected damage = (3.5) * (2/3) * (1) * (3.5) = 8.2

Now let’s add in re-rolls for either the shots or damage. Because these results are on a D6, we’re going to re-roll whenever the result is below a 4 (the average), which increases the expected result (we’ll talk about this in a moment). 50% of the time we’ll score a 4, 5, or 6, and the other 50% of the time we’ll re-roll the result, with an equal chance of a 1 through 6. That means our expected value on a D6 re-rolling is (1/2) * ((4+5+6)/3) + (1/2)*(3.5) = 2.5 + 1.75 = 4.25.

Re-rolling the number of shots

expected damage = (4.25) * (2/3) * (3.5) = 10.5

Re-rolling the damage

expected damage = (3.5) * (2/3) * (4.25) = 10.5

So these results are the same from an expected value standpoint. As we mentioned earlier however, in this case the sequencing matters for variance, if not expected value. An 11-sided die with faces running 2-12 has the same expected value as 2D6 but not the same variance – the latter will roll results of 6, 7, and 8 far more often, and in most games that predictability makes it a better option. Likewise, having five 2-damage shots produces less variance than having a single 10-damage shot. If our target has a 3+ save, the former will often produce 2-6 damage while the latter will either give us 10 or 0, and more often than not the result will be the latter. So when you have the option of re-rolling dice in a sequence, earlier rolls tend to produce better results – via lower variance – than later ones.

#### Things to Remember

• Driving rolls significantly impact the outcome of a particular attack; in general it is preferable to re-roll the number of shots over a particular gatekeeping roll.
• The most advantageous re-roll is when the result of a driving roll is less than the expected value, although the possibility of getting a negative result means that it may not be worthwhile to re-roll a value higher than one.
• The effect of an additive modifier (such as a +1 damage effect) on a driving roll or value (such as fixed damage) is highest when the value being modified is low.
• Multiplicative effects will influence probability equally when applied to damage or the number of shots, regardless of the original value. Bear in mind that in many games, excess damage is wasted, so this won’t always be true in practice –  for example, all other things being equal, a 10 shot 2 damage gun is mathematically identical to a 5 shot 4 damage gun, but if the models in the target unit have 2 or fewer wounds you’ll get higher performance out of the 10 shots than the 5 shots and against a single target the single shot has higher variance.

### Rolling Charge Distances in Warhammer

The other universal process in Warhammer 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.

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% Re-roll both dice 48% Re-roll one or both dice 58% Turn one die into a 6 92% 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 games 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 Intercessors about to shoot at something and my options are between Chaos Terminators and Chaos Marines, I can do some quick thinking to figure out how valuable it’ll be to shoot at each target. In this case say my Intercessors are more than 12″ away and moved. I can generally assume that of my 10 shots, about 6-7 will hit. Chaos marines are Toughness 4, so of my 6-7 half, half will wound, so 3-4. Then the Chaos marines get a 4+ save against my AP-1 bolt rifles and have a single wound each, so I can expect about half of those to be kills, giving me around 1-2 dead models. While comparatively the Terminators are likely to shrug off all but one of those unless my opponent rolls very 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 rolls you are likely to make and which ones are likely to fail
We already talked about charge distances above, perhaps 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 game plan 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 re-rolls for high-upside situations
There will be a lot of outcomes in your games that aren’t that big a deal. If you have a mechanic like Command Points that give you limited re-rolls (one per phase in 40k), you’re going to need to be judicious about how you use them. 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. Yes failing a 2+ save 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 re-rolls on 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.

Have any questions or feedback? Drop us a note in the comments below or email us at contact@goonhammer.com.