Wolf Claws: Math-Hammer 40K time

I take some time to look at the math behind the upcoming Wolf Claws to help you decide whether to re-roll to-hit or to-wound.

No matter how much I'm looking forward to modeling up and playing a Gears of War themed Guard army, I will always have a soft spot for my Space Wolves. I love their fluff, I love their playstyle and I'm going to love them more once they arrive in October.

Of some interest to me are the Wolf Claws, the Space Wolves version of Lightning Claws. According to several rumor sources, these will allow you to choose whether to re-roll the to-hit roll or the to-wound roll. The question is, which is better? Let's take a look:

Single Dice Rolls and Re-rolls

The image above shows for each dice roll the normal percentage, the additional chance by re-rolling it, and the total change of success. This is the basis for the following table, which compares all of the dice rolls in close combat:

Side-by-Side Comparisons

First, the terminology: when you see 3+r | 2+, that means 3+ to hit with a re-roll, followed by a 2+ to wound. So on the first row, you can see that if your roll is going to be a 3+, then a 2+, it's better for you to re-roll the 3+. Second, you can see that for rows where the requirement is the same (3+ to hit and to wound, for example), it makes no difference. That's simply rules of math: 3 x 2 is the same as 2 x 3.

Implications

Obviously, if you look at the chart for just a little while, you'll see a pattern emerging: in all of the comparisons, the re-roll is best utilized when it is used for the roll least likely to succeed. So given the choice between re-rolling a 4+ or a 2+, you want to re-roll the 4+.

Of lesser but still notable importance: how often are Wolf Claws advantageous over Lightning Claws and by how much? If you look at the chart, you'll see that of the 15 roll combinations there are six cases where they make a difference. And where the choice to re-roll to hit instead of to-wound is advantageous, the gain ranges from negligible (about 2.8%) to reasonable (about 13.9%).

Usage and Bottom Line

In gameplay, the Space Wolves player will want to do some quick thinking ahead before the dice are rolled to determine which roll will be more difficult and then declaring the re-roll to be that one. So if you will have to roll a 4+ to hit, then a 2+ to wound, re-roll the hit. But if you're rolling a 4+ to hit, then a 5+ to wound, re-roll the wound.

Wolf Claws are a fun and solid addition to the Space Wolves arsenal of weapons, and the re-roll choice they provide should always be used to re-roll the harder and I look forward to seeing them shred up enemies on the field of battle!

5 comments:

Dverning said...

Thanks for this. The results were fairly intuitive, but I hadn't sat down and done the actual spread.

I do see one item worth mentioning though: Where should a player spend the re-roll when the odds are even? A 4+/4+r may have the same statistics as a 4+r/4+, but what about probabilities? Say you have 10x dice...
4+/4+r rolls 10 to get 5 hits. Then rolls 5 to get 2.5 wounds. Re-roll 2.5 to get 1.25 more. Final average: 3.75. Total rolls: 17.5.
4+r/4+ rolls 10 to to get 5 hits. Then re-roll 5 to get 2.5 more. Then rolls 7.5 to wound. Final average: 3.75. Total rolls: 22.5.
So even though the averages are equal, the average number of rolls are not.

The more points of reference within a probability matrix, the more likely the system is to generate a result accurate to the statistical norms.
Put more simply, the more rolls you have the less likely you are to get "dice-screwed".

So the gambler will take the re-roll to wound, hoping for a lucky roll on his to-hit. But the calculating player will take it on his to-hit, knowing that it will give a better chance of getting the results he wants.

Cheers and hope that gave you some food for thought.

The_King_Elessar said...

Excellent post, great read, thanks.
To an extent it was clear anyway, but having the mathematical evidence to hand is always useful. :)

Colmarr said...

While you're obviously correct in saying that a re-roll is (generally) best used on the dice with the least chance of success, that ignores that fact that (presumably) one of the rolls has already been determined before the choice has to be made.

A failed to-hit roll is no longer a probability. It is a determined result (no damage) unless the re-roll is used. So for that roll, there is no viable choice other than to use the re-roll.

Ultimately, I can see this special ability working one of two ways:

1. the player chooses before making any rolls; or

2. the player chooses after making the to-hit rolls but before making wound rolls, and applies that choice to ALL attacks.

In scenario 1, your analysis applies and is very helpful. In scenario 2, the math is much more complicated than you have allowed for.

Dverning said...

@Colmarr: Space Wolf Codex, page 60, Wolf Claws: "the owner must choose at the beginning of each Close Combat phase."

So the analysis is Scenario 1 and thereby useful.

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