Wolf Standard: A Statistical Analysis

This article is here to determine the viability of paying 10 points for a wolf Standard. First, let's take a look at the exact wording of the item:

Once per game, a unit that includes a Wolf Standard may call upon the power of the wolf. For the duration of the next assault phase, all models in that unit may re roll any dice rolls of one.

Seems pretty simple, even seems like a bad wolf priest at first thought since you can only re roll 1's. BUT, you can re roll all dice thrown that assault phase...think of all of the die rolls you make in an assault phase. You can re roll 1's for difficult terrain on your charge move, to hit rolls, to wound rolls, armor saves, armor pen rolls, and sweeping advance just to name the main ones.

We're going to focus on the main rolls we care about...hitting and wounding. Instead of hitting with 50% of your attacks against models with a 4-8 WS, you'll hit with 58.33%. 8.33% doesn't seem too serious, especially for just one round...is it really worth 10 points? Well, let's say you have 8 Grey Hunters in the first round of combat when you use the banner. Because of counter-charge, you'll have 3A each regardless of if you or your opponent charged. That is 24 Attacks. Assuming you're going up against 4WS 4T opponents (and usually you are), you're going to get 12 hits and 6 wounds before armor saves without the banner. When you roll your 24A with the banner, however, math-hammer says 4 will be 1's. You just increased the number of attacks in your unit, effectively, by 4 since you get to roll 4 additional attack dice. Well, now that you've hit 14 times (12 initial hits, and 2 off of your 4 bonus attacks). With those 14 dice you'll get 2.33 1's and 7 rolls of 4+. The 2.33 re-rolls will net 1.17 wounds. This is a total of 8.17 wounds. Based on our hypothetical opponent, you would need to make 32.67 attacks to cause 8.17 wounds without the banner. Essentially, we got 8.67 additional attacks for one turn for 10 points. The more important thing to note is that we increased our wound output by 36.11% against 4WS 4T opponents for one turn with our basic attacks. So, 5 Grey Hunters would cause 5.1 wounds instead of 3.75 and 10 Grey Hunters would cause 10.2W instead of 7.5W.

Well, that's fine for when I'm fighting Marines and Orks, but what about IG, Eldar, Nids? What about when my Wolf Guard with a Power Fist is attached to the unit? What about when Ragnar is in the unit? Heck, that's a lot of calculations to have to do to figure out if we want to know Wolf Standard's viability across the board...or is it?

Here the a list of the percent chances of rolling a certain number or higher, right next to it I've put the percent chance of rolling that number or higher while re rolling 1's.

6+ 16.67% 19.44%
5+ 33.33% 38.89%
4+ 50.00% 58.33%
3+ 66.67% 77.78%
2+ 83.33% 97.22%

Notice a pattern? In every case, the ratio that the percent chance went up by was 1/6. In retrospect, it seems painfully obvious that this would be the case since we're going to be always re-rolling 1/6 of our dice. Well, that is good and fine for a single roll (and also good for pointing out that we'll be making 77.78% of our armor saves instead of 66.67%), but we're re rolling our "to hit" AND our "to wound" dice. Again, I first put the chance of causing a wound with a single attack and followed it by the chance to wound with a single attack re rolling all 1's. I've included the most common instances we'll come across, but take note that 4+ to hit and 3+ to wound is the same as 3+ to hit and 4+ to wound mathematically so I have only included it once here for brevity's(HA!) sake. If you try and do some of these calculations, please note that I originally took everything out to 10 significant figures and have rounded to 2 decimal places for ease of reading.

3+ to Hit and 3+ to Wound - 44.45% 60.50%
4+ to Hit and 2+ to Wound - 41.67% 56.71%
4+ to Hit and 3+ to Wound - 33.33% 45.37%
4+ to Hit and 4+ to Wound - 25.00% 34.03%
4+ to Hit and 5+ to Wound - 16.67% 22.68%
4+ to Hit and 6+ to Wound - 8.33% 11.34%

Once again we have a pattern across the board; you're getting a 36.11% increase in output when you incorporate both the "to hit" roll and the "to wound" roll.

In conclusion, we know that we're going to increase the success of all single rolls by a ratio of 16.7% and increase our total wounds caused by 36.1% for one turn with Wolf Standard, regardless of who is making the attacks or who they're attacking. Based on this, if you have any intention of your Grey Hunter squad getting in close combat, it should be more than worth the extra 10 points to increase your effectiveness by so much for one crucial turn...and all for the cost of a single Hunter-Killer missile.

If you guys enjoyed my article on Space Wolves tactics, then I'd suggest you check out Space Wolves Grey. I mentioned it last entry, but this site is everything you want out of a website dedicated to a particular army. They have painting and converting tutorials, really good tactics articles, army lists, pictures of their painted armies, and they're going to very soon start posting battle/tournament reports. In the mean time, next time I'll be posting some info on my Sapce Marine chapter, The Sons of Euclid.

Mark of the Wulfen or Power Weapon?

There has been a lot of discussion around the internet about having a Power Weapon vs. having Mark of the Wulfen in your Space Wolf squads. The discussion has, for the most part, come to the conclusion that power weapons are better against marines and the other MEQ's (marine equivalents), but that Mark of the Wulfen was better against weaker units like Orks and Imperial Guard. This particular enrty was inspired by an article written by Adam of Space Wolves Grey (http://space-wolves-grey.blogspot.com), the best site on the internet as far as I''m concerned for all things Space Wolves. In this case we're going to look at Grey Hunters. What makes this problem so "pure" is that they both cost the same (15 points), so there is no need to weigh cost/benefit when simply trying to decide which one is better...just benefit.

I like to use fractions of a "kill" when doing my comparisons. My basic method for comparison is to calculate the percentage of a single hit causing a wound (to include armor saves being taken) and multiplying that percentage by the number of attacks being made. Please understand that this is not a true scientific method for making comparisons, but it will give us a number that is very close to the true mathematical average...and is easy for the average 40k player to understand (I say "40k player" instead of "person" because the average 40k player is going to be smarter than the average person). Rending simply added an additional step in my calculations by taking into account that a certain percentage of wounds (aka those that caused a wound by rolling a 6) would be rending as opposed to a normal wound. This is why rending is particularly comparatively good vs. high toughness models...a larger percentage of your wounds caused will be rending (or 100% of them if you need a 6 to wound!).

Here is the average number of wounds caused by a Grey Hunter with a power weapon in the first round of combat. Note that these calculations include the fact that the opponent won't get an armor save due to the power weapon.

Power Weapon 3 Attacks
Guard Kills: 1.33
Ork Kills: 0.75
Marine Kills: 0.75

Here are the numbers for wounds caused by a Grey Hunter with the Mark of Wulfen in the first round of combat. Take note that I've included the rending bonus by the method I mentioned earlier.

MoW Attacks (with rending)
Guard Kills:
3A: 1.00
4A: 1.33
5A: 1.67
6A: 2.00
7A: 2.33
8A: 2.67
1.83 Avg Kills

Ork Kills:
3A: 0.67
4A: 0.89
5A: 1.11
6A: 1.33
7A: 1.56
8A: 1.78
1.22 Avg Kills

Marine Kills:
3A: 0.42
4A: 0.56
5A: 0.69
6A: 0.83
7A: 0.97
8A: 1.11
0.76 Avg Kills

It is important to note that one thing the traditional "mathhammer" method of making decisions doesn't take into account is that 6 attacks that hit on a 3+ is not the same as 8 attacks that hit on a 4+ Although they both "average out" to 4 hits using the traditional method of (% chance of success) * (# of attempts), the true statistical average will be different for both of them. It also doesn't take into account that there is a 0% chance of the former scoring 7 or 8 hits. That being said, our method here is still a good tool (albeit not a perfect one) for making decisions and we are able to say with some certainty that Mark of the Wulfen is better than a power weapon in each of these three cases. It is decidedly EVEN better against Plague Marines, Marine Bikes, or even Carnifexes. It is NOT better against striking scorpions or incubi (elite power armored Dark Eldar for those of you who weren't playing 40k in 2001).

Next week we'll take a look at the true effectiveness of the Wolf Standard, another option for Grey Hunters. In the mean time, you comments are appreciated. Is there anything out there you'd like to see a statistical analysis on?

Welcome!!!

This is a blog that has been on the tip of my mind for some time....

I am Euclid, a player of Games-Workshop games for over 15 years now. I am also an amateur statistician, having a bachelor's degree in finance and mathematics. This blog is going to be dedicated to two main aspects:

1) The rise and fall of my space marine chapter, "The Sons of Euclid" This will include painting progress, army lists, battle reports, and tournament reports.

2) A mathematical and statistical analysis of our decisions in Warhammer 40k to include the mathematical viability of different units and upgrades in list building as well as decision making during the game itself. 40k is a game ruled by numbers...be those the one's we roll, the points we spend, the characteristics we use, the distance we move and shoot....we can maximize our play by knowing the formulas that are constantly being processed with every move we make in 40k. We can choose to learn them and go into our games with that knowledge, or play blindly to what is happening around us.

Welcome