Dice Theory! A basic introduction

What is it?

Dice theory is about calculating basic probabilities which thus lets you estimate the success and fail rates of model performance, equipment and abilities. This then lets you judge their performance against other models, both your own and the enemies. From there you can use it to help you build army lists, make choices on the battlefield and evaluate results of battles you take part in.

Is it hard?

No. This is fairly basic maths that (with a bit of a reminder) most people can understand pretty easily. Yes if you really want you can take it far further with complex statistics and comparison methods, but for most users the best results are going to be just looking at the basic results. Though if you are keen by all means take it further!

Lets get started!

Note for this discussion the / line is being used to show a fraction not a division sign (although note that to put any fraction into normal numbers you simply treat it as a division, just be sure that you do any division before any multiplication when doing your sums)

The first core concept to understand is how to represent rolling in maths. The Warhammer games are easy in this regard because the random elements introduced by dice rolling are all done on the good old D6 – the six sided dice. Therefore you know that in pretty much all cases the probability of anything happening is going to be out of six.

So lets look at a model – the Daughter of Khaine Witch Aelf. This model has a To Hit value of 3+, which means any roll on a D6 which is a 3 or higher will hit.

A 3+ means that a roll of 3, 4, 5 and 6 on the dice will succeed. So that is a total of 4 possible rolls from a dice with 6 sides. We can write that as 4/6 and can also be divided to give 0.6666 to infinity (it never ends) which can be simplified to 0.67 (which is a perfectly fine level of accuracy for dice theory)

So if one recalls that a half is 1/2 = 0.5 we can see that for a “To Hit” of 3+ a model should be hitting more than half of the time when the dice rolls. It won’t be every time, but its a pretty good success rate.
However landing a hit is only part of an attack, you also have to roll to wound. A Witch Aelf has a To Wound roll of a 4+, so a roll of 4, 5 and 6 will land a wound. That is 3/6, or 0.5 (yes same as 1/2 – a half). Now because to wound depends first on the chance of the model hitting we have to combine the two results.

To Hit – multiplied by – To Wound = Number of wounds scored.

4/6 * 3/6 = 0.33
0.67*0.5 = 0.34 – Note there is a tiny difference here because we used the rounded value instead of the infinite for 4/6. In terms of dice rolling this difference is negligible.

So now you can see that for a single Witch Aelf making a single attack, it has under a half chance of landing a wound each time. In fact doing the maths with the proper values its exactly equal to 1/3.
Now we can take that understanding and improve it further. Lets say we’ve got 10 Witch Aelves in a single unit. They each have 2 attacks, 3+ hit and 4+ wound and deal 1 damage with each hit. Also there is one leader in the group who has a +1 to hit. Let us assume that you get this unit into close combat and that all 10 models are in range of the enemy.

We thus have:
9 models make 2 attacks each with a To Hit of 3+.
1 model making 2 attacks with a To Hit of 2+ (2, 3, 4, 5 and 6)

We can write that as:
The attacks made by the regular models*to hit chance + attacks made by the leader*to hit chance = total attacks
(9*2*4/6) + (1*2*5/6) = 13.67
12 + 1.67 = 13.67

So from 10 Witch Aelves we can estimate that they will make 13.67 attacks. From there we can work out how many times they will wound from those attacks. Because they all wound on the same 4+ score this is even easier and is simply

Number of attacks*to hit chance = number of wounds
13.67*3/6 = 6.83

So from 10 Witch Aelves you can expect to make 6.83 wounds (which you could round up to 7 if you wish). As 10 Witch aelves make 20 attacks total we can see that it matches our earlier maths of  being around 1/3rd of the attacks

1/3*20 = 6.67

The leaders bonus makes a little difference, but not a huge amount in this comparison, but likely pushes it closer to a full 7.

 

Now in this test these Aelves were not attacking anything, but if we give them an opponent – say a nasty Deamonette of Slaanesh – we can see how things might fare.

A Deamonette has a basic save of 5+ (works on a 5 and 6), and we are making a (rounded up) 7 attacks against them. So that would be:

Number of wounds*save chance
7*2/6 = 2.3

So we can expect them to make around 2 saves, which would mean they’d lose 5 hit points; which as each Deamonette has only 1 wound, means they would lose 5 models.

In practical terms we now know that if we take our unit of 10 Witch Aelves and attack a unit of 30 Deamonettes, we are only going to have a chance to kill around 5 of them in any one turn of combat, which is not that many from such a large unit. In contrast, if we were to charge a unit of 10 Deamonettes we could expect to take out around half the unit!

 

Whilst experience on the tabletop can also teach these very same lessons, a little bit of maths theory can let you better understand these concepts in less complicated manner.
Furthermore its not beholden to wild-chance that real dice will give; for example you could attack 30 Deamonettes with your 10 Witch Aelves and kill 15 of them with some really super lucky dice rolls for you and some bad saving from your opponent. An experience like that might make you think your Witch Aelves are superpowered and thus charge them into Deamonettes (and other similar units) like that again, only to get crushed when you kill a more “normal” number of 5 and they make a far superior return attack.
It can also help a lot when you want to compare things, such as different unit types or weapon choices on the same unit. You can test out what the wounds will likely be, but also see how different special effects might or might not become more critical to the importance of a unit The next article covers this in more detail by comparing the Witch Aelves with duel blades and Witch Aelves with bucklers.

Remember Dice Theory is not replacing gameplay and actual experience. It makes assumptions about the game situation and often you might compare things at the extreme ends (eg comparing 10 witch aelves and 30 witch aelves) to see the patterns or differences in the units. Furthermore its not taking into account positioning, range, cover etc…All essential gameplay elements that contribute to an overall success. Indeed in this test we’ve assumed that you’d get all 10 models in the unit into close combat – in a real game you might lose several on the way and might also not be able to move all of them into position to attack. However if you know that about 1/3rd of your hits are going to cause wounds and most of those will kill you can at least plan with that in mind accordingly.

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