One of the most common calculations we do for Hammer of Math is trying to figure out how many wounds a weapon will deal on average, given some number of attacks, the toughness and save characteristic of the target, and any special rules being applied. In today’s Hammer of Math we’re going to talk about how those calculations are done and how you can replicate them on your own to do your own quick calculations.
Expected Value
In Statistics, the Expected Value is the mean of the possible values a variable or series of outcomes can take, weighted by the probability of those outcomes. In other words, a probability-weighted average. Let’s look at a couple of quick examples:
1. Rolling a D3.
Let’s start with an easy example – rolling a D3. We want to get the expected value of a D3, so we look at the weighted mean of the results. Each result has a 1/3 chance of happening, so our expected value calculation looks like this:
(1/3) * 1 + (1/3) * 2 + (1/3) * 3 = 1/3 + 2/3 + 3/3 = 6/3 = 2
So the Expected value of a D3 is 2. That makes sense intuitively – you know that you’re going to roll 3s as often as you roll 1s, and if you roll a bunch of D3s, you will generally count on them averaging out to a bunch of 2s. That’s because the larger the pool of dice, the more likely we’ll get a result close to the expected value. Let’s take a look at that in action.
2. Rolling a D6.
If we look at a single D6, we get:
(1/6) + (2/6) + (3/6) + (4/6) + (5/6) + (6/6) = 21/6 = 3.5
Now you can’t actually roll a 3.5 on a D6, but half the time you’ll be above that and the other half you’ll be below that. If we add another D6 and roll 2D6, the expected value is just the sum of those values, i.e. 3.5 + 3.5 = 7. That’s both the expected value and the most common value if you roll two dice, and about 1 in 6 rolls will get you a total of 7.
The Expected Value of an Attack Sequence
When we start figuring out the value of an attack sequence, we start by thinking about what the desired outcome is. And most of the time, what we’re looking to do is deliver unsaved wounds – and damage – to the target. An attack sequence consists of four key parts:
- Hit roll
- Wound roll
- Save roll
- Damage
And so a successful attack is one that hits, wounds, and goes unsaved, dealing damage. In statistics, the probability of multiple events occurring in sequence is usually done by calculating the conditional probability but the good news is that with dice, those events are always statistically independent, i.e. each roll of a die has no impact on the next, so the chances of two dice events happening concurrently is just the product of their probabilities.
So the chance of something hitting and wounding is just the chance of hitting multiplied by the chance of wounding. Likewise, the chance of an attack going unsaved is the probability that, after applying an AP modifiers, the save roll will fail, or 1 – P(save). Putting this all together, let’s model one space marine intercessor shooting a bolt rifle at another.
- To Hit: A single bolt rifle attack has BS3+, so will hit on four out of six possible dice values – 3, 4, 5, and 6. So we can use 4/6 = 2/3 for our probability to hit.
- To Wound:Â Bolt rifles are Strength 4, and marines are Toughness 4, so we’ll wound on a 4+, or three out of six possible values on a D6 (4,5,6). That means we can use 1/2 for our probability to wound.
- Unsaved:Â Space Marines have a 3+ armour save and bolt rifles are AP-1, reducing them to a 4+ save. That means they’ll save on 3 out of 6 values on a D6 (4,5,6), so our chances of an unsaved wound are 1 – 1/2 = 1/2.
So our chance of delivering an unsaved wound with a bolt rifle is going to be the product of those three values, or:
(2/3) * (1/2) * (1/2) = 2/12 = 1/6
This means for every six shots we shoot, we should expect to get one wound through on the target, more or less. In fact, this is often what we’re doing when we do expected value calculations – we’re looking at larger pools of dice. If, instead of shooting one Intercessor’s bolt rifle, we had a squad of five shooting using the unit’s new Target Elimination ability, which gives them +2 Attacks when shooting at a single enemy unit. Each Bolt Rifle starts with 2 attacks, so adding in 2 each for Target Elimination brings them to 4. So we can multiply our output by the number of shots, which in this case will be 4 per model, or 5 * 4 = 20.
Thus we get:
20 * (2/3) * (1/2) * (1/2) = 40/6 = 6.67
So on average a unit of five Intercessors unloading into another should expect to do around 6-7 wounds to that target, taking out around 3 models. Is this always going to happen? No, but it gives a good idea of what to expect if we’re throwing dice that way. We may get more wounds than that, but we definitely shouldn’t count on them, and we could definitely end up with fewer.
Applying Modifiers
Applying Modifiers here is easy – simply modify the probability of the hit roll, wound roll, or save probability as needed. So if we had +1 to hit with the bolt rifles from say, remaining stationary with their [HEAVY] ability, we’d be hitting on a 2+, or with five out of the six values on a D6, increasing our hit odds to 5/6. If we improved our AP by 1, we’d increase our probability of an unsaved wound – the chance of a success would drop to 1/3 (a 5 or 6 on a D6), and 1 – 1/3 = 2/3. And if we got +1 to wound from something like Oath of Moment, we’d improve those odds as well, going from a 1/2 chance to wound to 2/3 as w/ed wound on a 3, 4, 5, or 6. On that note, let’s talk about Oath of Moment.
Applying Re-rolls
One of the most common modifiers in the game is re-rolls. When you re-roll a result, you’re improving the probability by adding an additional chance of success after a failure. The way to do this is by adding a probability for that second chance. If your hit roll is a P(4+) = 1/2 and you have full re-rolls, then the half of the time you miss you’ll get to make another attempt, this time again with a (1/2) chance of success. So your odds look like:
P(4+ with re-rolls) = 1/2 + (1/2)*(1/2) = 1/2 + 1/4 = 3/4
Or more generally:
P(success) + P(failure with a re-roll) * P(success)
If we give our Intercessors full re-rolls to hit, their output against that other unit of Intercessors looks like this:
40 * [(2/3) + (1/3)*(2/3)] * 1/2 * 1/2 = 40 * 8/9 * 1/4 = 80/9 = 8.9
These values don’t have to be equal, as you might have already guessed. If you only re-roll 1s to hit, then instead of re-rolling on the 1/2 of attacks where you miss, you’ll only re-roll on the 1/6 of times you roll a 1 to hit. In other words:
P(4+ re-rolling 1s) = 1/2 + (1/6)*(1/2) = 1/2 + 1/12 = 7/12
(generally speaking, adding re-rolls of 1s is also the same as just multiplying the odds by 7/6 so you can shortcut it that way). Going back to our Intercessors example, we get:
40 * [(2/3) + (1/6)*(2/3)] * 1/2 * 1/2 = 40 * 14/18 * 1/4 = 140/18 = 7.8
Note that single die re-rolls are much more complicated, and generally better done with a simulation tool rather than simple expected value calculations, as they add a lot more complexity.
Sustained Hits
Sustained Hits affects the number of hits you’ll be scoring, bypassing the hit roll to generate extra hits in the sequence. The way I tend to do this is by adding in extra hits – (1/6) * the number of attacks. This ends up splitting the calculation to an intermediate step, but you can combine it all into a single step, where your math shifts from attacks * P(hit) * P(wound) * P(unsaved) to hits * P(wound) * P(unsaved), where hits = attacks * P(hit) + Attacks * (1/6). Feel free to adjust based on your updated odds if you have crits on 5+ .
For our Intercessors, giving them Sustained Hits looks like this:
Attacks * [P(hit) + 1/6] * P(wound) * P(unsaved)
which is 40 * [2/3 + 1/6] * (1/2) * (1/2) = 40 * 5/6 * 1/4 = 50/6 = 8.3
So adding Sustained hits
Lethal Hits
On the other hand, Lethal Hits affects your wound probabilities in a similar function, though by bypassing the wound roll. This will again affect your sequence, forcing you to calculate the expected wound counts. In this case, 1/6 of your attacks will be lethal hits, so you subtract those out from the sequence and assign them a different wound probability – these are guaranteed wounds, so their wound probability P(wound) = 1. For the rest you still have to roll. As an example, if our Intercessors had Lethal Hits, their attacks look like this:
Attacks * [P(Nonlethal Hit) * P(wound) + P(Lethal) * 1)] * P(unsaved)
which is 40 * [(1/2) * (1/2) + (1/6) * 1)] * (1/2) =Â 40 * 1/8 + 40 * 1/12 = 5 + 3.3 = 8.3
And this also demonstrates nicely that for those times where we need a 4+ to wound, Lethal Hits and Sustained Hits are identical in terms of their expected benefit.
Applying Damage
We haven’t talked about damage yet and how it applies. This is where things get tricky, and our targets matter a lot more. pretty much any time we have 1-damage weapons or going into a big multiwound unit like a Knight we can just go with this flat calculation and add the damage characteristic as another value in our product. So if we had 2-damage Heavy Intercessors shooting a knight, our value would be:
Attacks * P(hit) * P(wound) * P(unsaved) * Damage Characteristic
to get the final expected damage value. This is good for big targets but when and how we use damage will depend on the way we’re calculating damage. If we were shooting Intercessors then rather than look at raw wound output, we’d likely want to look at dead models, and so for 1-damage weapons we’d divide the value by 2, and for 2+ damage weapons we’d just leave off the damage calculation to get a number of expected kills. This also helps us understand what things look like if the excess damage was lost. Likewise, against something like Terminators it will take two unsaved wounds per Terminator so we’d again halve the final result to get effective kills.
Final Thoughts – How to Use This
This should be everything you need to get started. As we mentioned earlier, picking targets is a huge part of this – you have to know what you’re attacking in order to get those wound and save values, as well as to know how damage will be applied and matter. In terms of “how do I use this?” well the most common way is going to be comparing two different weapon options. We’ve done that countless times in this column, looking at how different profile options compare or understanding things like melta vs. plasma. You can do similar calculations on your own for your own unit choices, comparing the expected value outcomes of each option to determine which is the best for a given situation or target.
Have any questions or feedback? Drop us a note in the comments below or email us at contact@goonhammer.com. Want articles like this linked in your inbox every Monday morning? Sign up for our newsletter. And don’t forget that you can support us on Patreon for backer rewards like early video content, Administratum access, an ad-free experience on our website and more.
