Spell Generation Probability in Warhammer

So I thought I’d start things off with a mathsy post. Recently I’ve been discussing the new High Elves army book with someone on reddit. After replying to a post of his a few months ago, he PM’d me with an enormous wall of text asking for advice. He bigged up my ego to begin with so I was happy enough to oblige.

He was singing high praises of Alarielle the Radiant and specifically High Magic, loving the Lore attribute which gives you +1 to the ward save of the wizard and his unit or a 6+ ward save if they do not already have one. Alarielle works in a similar way to the Skaven Grey Seers in that you can choose a number of spells from multiple lores and then randomise that many spells. She gets to choose 4 spells from the lores of Life, Light and High Magics.

In the list that this guy proposed, he had opted not to take Shadow magic which I think is particularly devastating with the High Elves and claimed that it was because he couldn’t guarantee getting Occam’s Mindrazor. (I know. How awful is that name?)

As it is a really simple process to generate spells, I worked out how likely you might be to generate spells.

I’m going to ignore the possibility of multiple wizards on the same lore because a) it starts to get more complicated and b) I’d have to go and re-read the rules on how that worked anyway.

For those that have little or no experience in Warhammer Fantasy, there are eight lores of magic within the main rulebook and some armies have their own. In the main rulebook and the 8th edition rulebooks, each lore has one “signature” spell and six other spells. When generating spells, a wizard will roll a number of dice equal to his wizard level and will know the corresponding spells for the battle. A wizard may choose to substitute any single spell for the signature spell and in the event of doubles, triples or a quadruple, he may choose any other spell with the second, third or fourth instance of that number. Clear as mud? Excellent. In practice, it’s pretty simple.

I’m going to ignore wanting to obtain the signature spell because that is obvious; if you want it, you can have it. Let’s say that for any particular lore, you want to obtain one specific spell. In order to get that spell, you need to have rolled it or rolled a double of another number.

As a level 1 wizard, this is trivially simple, you must roll exactly the number you want. 1/6 chance.

As a level 2 wizard, this becomes more complicated as you have two chances to roll the number + the chance of rolling a double. So of the 6^2 = 36 combinations, 6 of them are doubles, 6 of them contain the number on the first dice and 6 of them contain the number on the second dice. There is obviously some overlap here so we must be sure not to double count. For simplicity’s sake, let’s say we’re looking for spell number 6 (generally the most powerful of the lore). Rolls in which we will get spell 6 are (1,1),(2,2),(3,3),(4,4),(5,5),(6,6) and (6,1),(6,2),(6,3),(6,4)(6,5),(6,6) and (1,6),(2,6),(3,6),(4,6),(5,6),(6,6). We’ve counted (6,6) three times so we have 6/36 + 6/36 + 6/36 – 2/36 which is a 4/9 chance.

As a level 3 wizard, this is more complicated still. So much so that it makes it simpler to look at the rolls in which we cannot get the spell we wish. Again, let’s take spell number 6. Ignoring order we could roll (1,2,3),(1,2,4),(1,2,5),(1,3,4)(1,3,5),(1,4,5) or (2,3,4),(2,3,5),(2,4,5) or (3,4,5). This is 10 ways of rolling the three dice. Each of these has 3! = 6 possible orderings. That is 60 ways to roll the dice out of the 6^3 = 216 ways. This is a 60/216 = 5/18 chance not to get the spell. So a 13/18 chance.

As a level 4 wizard, we have fewer ways of rolling to avoid the 6. (1,2,3,4),(1,2,3,5),(1,2,4,5),(1,3,4,5),(2,3,4,5). This is 5 ways and each has 4! = 24 possible orderings. This makes 120 ways to not roll the spell out of the 6^4 = 1296 ways. So a 49/54 chance.

If the game were to go to wizards rolling for 5 spells, there is exactly one roll (1,2,3,4,5) with 5! orderings. Rolling your spell would be 7656/7776 chance.

Obviously at 6, it is impossible not to gain access to your spell.

So this was probably all pretty confusing but the upshot is this table.

Getting multiple desired spells becomes much more complicated and we might need some real maths to solve that so I’ll hold off for today. Hope you enjoyed!