This week’s Hammer of Math bows before the Blood God and examines the probabilities behind the Blessings of Khorne army rule for World Eaters.
Say what you will about Games Workshop, but they certainly know how to hype up a product. Faction reveal after faction reveal has had players excited for the possibilities of the new edition, and the promise of a complete reboot means anything could happen. Last week Warhammer Community dropped rules for how the World Eaters would play, and the math is almost as interesting as Angron is terrifying.
Let’s start by looking at the new army rule:
The core mechanic for World Eaters has changed – the Blood Tithe has been replaced by a new rule called Blessings of Khorne. This rule has you roll 8 dice at the start of each battle round; after you do, you can then activate up to two Blessings from the list below (or possibly resurrect Angron). Which abilities you choose depends on the available dice rolls, and different combinations of rolls can give you different effects.
First thing’s first: With 8 dice you are guaranteed to get either one triple or two doubles (this is due to a principle called the pigeonhole principle, if you’re interested). You can further improve the probability of success by putting a Khorne Berzerker unit with an Icon of Khorne within range of an objective marker. Each icon in range allows one dice to be re-rolled.
Rage-Fuelled Invigoration / Wrathful Devotion
Since eight dice are being rolled and there are only six faces, a double is guaranteed. What about the probability that two doubles will occur? If we drop the two dice that are required for the first double, we have a pool of six dice remaining. The first dice can be any arbitrary value. The chance that the second dice doesn’t match the first is 5/6. The chance that the third doesn’t match either of the first two is 4/6, fourth not matching is 3/6, fifth is 2/6, and the chance that the sixth doesn’t match is 1/6. So the chance that none of the dice match is the product of the probabilities, or 1.5%. Since any other outcome produces a second double, that means that the chance that there are two doubles is 98.5%.
The math gets interesting when we have to calculate the probability of a double whose values are equal to or greater than a target number. In order to do this I put together an AnyDice script. The way the script works is that you provide a target number and a pool of dice, and then AnyDice looks over all the sequences and counts the number of instances where the desired double or triple occurs. In the case of Martial Excellence this translates to the probability of 8D6 generating a double of 3+ to be 94.3%, which seems reasonable given the sheer number of dice available.
With Total Carnage the probability of getting a double 5+ on 8D6 is 66.8%, but the chance of getting any triple is 70.7%. Together the script says the net probability of Total Carnage is 95.4%, which is actually more likely than Martial Excellence.
With the same option to leverage a triple as Total Carnage, the probability of getting a result to enable Warp Blades is 90.3%.
When we start looking at single value goals we can leverage our good friend binomial probability, which is the chance of a given number of successes at a fixed probability over a set number of trials. Both the script and the linked calculator indicate that the probability of getting a double 6 is 39.5%, which is helpful validation for the script. Add in the probability of getting a triple 4+ and the cumulative probability to enable Unbridled Bloodlust is 62.8%.
Reborn in Blood
In what can only be the perfect example of someone who is simply Too Angry To Die, Angron has an ability named Reborn in Blood which allows him to return to Reserves (and subsequently Deep Strike) after he dies by using three 6’s from the pool. As far as I can tell this is not considered a Blessing of Khorne so the player can still activate two Blessings if they have the dice. The chance of getting the necessary result is 13.5%. This goes up to 17.8% if one dice is re-rolled, and 22.5% if two dice are re-rolled courtesy of an Icon of Khorne. The synergy between Berzerkers and Blessings of Khorne is pretty considerable.
Blood for the Blood God
The numbers indicate that while the Blessings of Khorne mechanic does have some variability to it, the chance of getting a desired effect is pretty high. You can further enhance this probability through multiple Icons of Khorne, as each Icon in range of an objective effectively increases the pool of dice by one as you re-roll an undesirable result. While we don’t have the full picture to know how well this ability to work in teh full game, it certainly looks interesting and full of iron-tinged flavor.
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