An Introduction to the Statistics of Warhammer 40k Shooting
Building a Warhammer 40k that is effective at winning games is difficult to do. One of the reasons this is difficult is that the point value of a particular unit or upgrade does not give you the statistical effectiveness of the unit or upgrade. I am writing this article to help new players, or players who haven’t delved into the probabilities of Warhammer 40k, see what the average result of some of the actions in Warhammer 40k. This article will go into some of the details of the Space Marine codex and discuss the probabilities of causing unsaved wounds/damage to vehicles of certain units and upgrades and compare them to the point cost of the unit or upgrade. This article only goes into the effectiveness of the units and upgrades the moment they initiate a shooting attack or charge into combat. This article does not describe strategies that lead up to these moments. These statistics should be used when planning strategies so that you can effectively use your army. For example, a marine with a heavy bolter will cause 1.67 unsaved wounds, on average (when I give these numbers assume it’s the average result unless is say otherwise), to a tau fire warrior (toughness 3 and armour save 4+) while the auto cannon on a predator will only cause 1.11 unsaved wounds on average. The heavy bolter will cause 0.67 wounds to a toughness 6 creature while an autocannon will cause 0.89 wounds. The zone of effectiveness of these weapons should be known and this article will help show some of these zones of effectiveness.
I personally collect Space Marines and going to go through some examples that can be extended to other races. These first couple tables show the chance to wound when shooting at infantry and the chance to cause damage (glance or penetration) when shooting at vehicles. These tables are easily made. For the “to wound” probability table, just subtract the toughness of the model from the strength of the weapon, add three to this value, if the value is above 5 decrease it to 5, if the value is equal to zero add 1 to it, if the value is below 0 you increase the value to 0, and then divide this value by 6 and you have the probability of wounding the model with that weapon strength. Lets go through an example: my bolter can shoot at a toughness 4 creature, a toughness 6 creature and a toughness 8 creature and I want to find the probability of wounding each creature with my bolter strength 4. First the toughness 4 creature, so toughness 4 minus strength 4 is equal to zero, then add three so we have 3+0=3, its below 5 and above zero so we just divide this number by 6 and get 50% (3/6) chance to wound. For the toughness 6 creature, 4-6 = -2, -2+3=1, and then it is in the correct range so the chance of wounding is (1/6) or 16.7%. For the toughness 8 creature, 4-8=-4, -4+3=-1, -1 which is below zero so we increase it to zero, 0/6= 0% a bolter cannot hurt toughness 8 creatures.
For the “to damage a vehicle” probability table you first add 6 to the strength of the weapon, subtract the armour of the vehicle from this calculated value, if the value is less than zero then the strength of the weapon cannot hurt that armour, if the value is not negative then add one to this number and divide this number by 6 and you get the chance that weapon will cause damage to that armour value. Lets look at the probability of a lasscannon hurting an armour value 11, 13 and 14. For armour value 11, 9+6=15, 15-11=4, 4+1=5, 5/6 chance of hurting the armour value 11 or 83%. For armour value 13, 9+6=15, 15-13=2, 2+1= 3/6 or 50% chance of hurting the armour value 13. And finally for armour value 14, 9+6=15, 15-14=1, 1+1=2, 2/6 or 33% chance of hurting the armour value 14. Probabilities of meta guns and weapons that use two dice to determine penetration use a different set of rules to find the probabilities of damage (or success). To find this probability you first find the minimum number added to the strength of the weapon that will result in a glance, then find all the combinations to get a number less than the minimum number and subtract this number from 36 or just find all the combinations that result in a number equal to or greater than this minimum value and divide this number by 36 and you get the probability of damaging this armour value. Lets look at a melta gun shot at half range against a land raider, the melta gun shot needs 6 or more to do damage to the land raider (8+6=14), there are 10 combinations of two dice to get a number less than 6 so we subtract this from 36 and get 26, and finally divide it 26/36= 72% chance to do damage to a land raider at half range.
These damage probabilities are not all that matter when choosing which unit or upgrade to buy. The amount of shots, the ballistic skill of the wielder and the save of the targeted model also affect the effectiveness of the shooting. Let’s look at how to find the average amount of unsaved wounds a space marine with one bolter shot will inflict on an Orc and on a chaos space marine. First we check the chance the space marine has to hit his target, 4/6 chance for BS 4. Then we check the chance to wound the target 3/6 in both cases. Then we check the chance to fail their armour save, the Orc has a 100% chance to fail the armour save and the chaos space marine has a 2/6 chance to fail the armour save. Now to calculate the average unsaved wounds: for the Orc= (4/6)*(3/6)*(1)=0.33 and for the chaos space marine=(4/6)*(3/6)*(2/6)=0.11.
EQUATION TO REMEBER: (Chance to hit)*(Chance to wound)*(Chance fail save)*(Number of attacks)=Average number of unsaved wounds.
From these numbers we can see that it should take, on average, 3 bolter shots from a marine to kill an Orc and it should take roughly 9 bolter shots to kill a chaos space marine. An Orc is worth roughly 6 points (I think) and a chaos space marine is worth about 15 points. From the numbers it looks like the chaos space marine can take 3 times the punishment as an Orc can. I think the reason why the chaos space marine isn’t quite three times the cost of an Orc, is because plasma weaponry and low ap weapons kill chaos space marines with the exact same average unsaved wounds as Orcs and Orcs have more attacks in close combat than Chaos space marines. But this article is not about discussing orcs and Chaos space marine differences, so lets move on.
Average damage in close combat is much more complicated to determine than shooting averages. I believe that you can estimate the average result using a similar method to the shooting averages but since the distribution of combat effectiveness is not even through almost all units in the game some casualties from earlier initiatives have a chance to kill the sergeant with the power fist or nob with the power klaw even if the average amount of unsaved wounds is less than the defenders squad size. For example if a squad of 10 space marines with a sergeant w/ power fist get charged by a squad of Khorn berserkers, my sergeant has a chance of dying if the Khorn berserkers cause 10 or more wounds. This chance of significantly decreasing the combat effectiveness of the squad seems to effectively branch the results into two possibilities, one where my sergeant lives and does his damage and the other where he dies and doesn’t. I believe if you where to multiply the chance the sergeant has to live by the damage he can inflict you should get a pretty good value for the average amount of damage he will inflict. Now this adds to the calculations of these scenarios significantly and I personally would leave these calculations to the computers.
I know some people don’t believe in statistics or math and feel it’s too abstract to apply to their games. Well I want to help decrease the abstraction by going through a 600 point battle I had against my friend. I have the video up on Youtube, and I will go though the average result of each action taken and compare them with the result in the game. I categorized the damage done to vehicles as permanent and non permanent to help show some averages because the chance to get one of the specific results on the table is very low most of the time. (When you roll on the table without modifiers you have the same chance to get all the results)
Dark Angles Turn 1
The snipers in the buildings shot at my rhino and immobilized it. They are Dark Angles scouts so they have BS 4 so 4/6 chance to hit. Snipers need a 6 and then a 3+ to do damage to armour 11. Chance to glance = (1/6)*(1/3)=1/18, chance to penetrate = (1/6)*(1/3)=1/18. Glance has 2/6 chance to cause permanent damage, penetrate has 4/6 chance to cause permanent damage. 1/18*2/6 = average permanent damage done by glance and 1/18 * 4/6 = average permanent damage done by penetration. We add these two statistics together because they don’t overlap, and we get 0.0555... for one sniper shot. Lets multiply this by 5 to see what the average permanent damage result the squad causes, and we get roughly 0.28 so it should cause one permanent damage result ever four rounds of shooting. Now I didn’t look at the average non permanent damage the shooting should do to save time. If I did calculate this number then you could see the average results the shooting should have done, cause shooting can cause both kinds of damage and thus you should have averages for both in both cases. I will only really look at either or later on just to save time.
That’s all that really happened DA turn 1.
Ultramarines turn 1
I had my rhino and marines fire at the speeder and managed to shake it. It ended up being four bolter (two from rhino storm bolter and 2 from the 2 marines inside) shots as it was close to 24” range. Chance of non permanent damage is (1/6)*(4/6). So we get the average result (4/6)*(1/6)*(4/6)*4 shots = 0.3. It seems that my opponent’s luck early on was matched by my own.
I then fired my two speeders at his own rhino. Two krak missiles and 6 heavy bolter shots were fired. Lets find the average result for each separate shot type and add them together. Chance to hit 4/6. Chance to do non permanent damage (1/6)*(4/6)+(3/6)*(2/6). And the average non permanent damage result= (4/6)*( (1/6)*(4/6)+(3/6)*(2/6))*2shots=0.37. And now the 6 heavy bolters average result is (4/6)*((1/6)*(4/6))*6=0.44. Add these averages together and we get 0.81. It seems like the stun I get on his rhino is actually pretty likely to happen.(This is a reminder that the 0.81 is not a percentage it’s an average. If your confused I suggest going to Wikipedia and looking up their statistics section)
Dark Angles turn 2
The Snipers succeeded in doing permanent damage to my rhino a second time. Again this is supposed to happen once every four shooting phases or so, so I feel this can be classified as lucky.
Ultramarines turn 2
Now there was a lot of shooting at the land speeder this turn. For the 7 bolter marines + sergeant bolt pistol, average damage (of any form) result = (4/6)*(1/6) *15=1.67. Leader BS 5 bolt pistol average damage (5/6)*(1/6)=0.14 . 9 man bolt pistol scouts BS 3 average damage = (3/6)*(1/6)*9= 0.75 And the sergeant of the Scouts squad BS 4 average damage = (4/6)*(1/6) =0.11. All these averages added together gets 2.67 average damage results. This average shows that something really should have happened to the speeder here. This was bad luck for me but I was having a good time so it was ok.
I shot the speeders again and got a non permanent damage
result, which is close to the average.
My 10 man squad of scouts assaulted the speeder with their krak grenades and cause it to explode. Average lethal damage result(Which is just permanent damage excluding the weapon destroyed result) = (1/6)*((1/6)*(1/6)+(2/6)*(3/6))*10=0.32. Now killing the speeder with the assault was lucky in my favour but this seemed to even out my bad luck with my bolters. But the speeder exploded and killed 3 scouts. The explosion got 10 hits on my scouts at strength 3, what is the average unsaved wounds for this event? 1*(2/6)*(3/6)*10=1.67 . It seems like one to two scouts would die on average. The three scout’s dying is a little bit off average.
Dark Angles turn 3
The passengers of the rhino got out and shot at my speeders causing one shaken result. This is from 1 melta gun shot not at half range and 7 bolter shots from the rest of the marines and the 2 storm bolter shots from the leader. Melta non permanent=(4/6)*((1/6)*(3/6)+(4/6)*(1/6))=0.13 . Bolter shots non permanent = (4/6)*(1/6)*(4/6)*7=0.52. And leader storm bolter non permanent=(5/6)*(1/6)*(4/6)*2=0.19. Added together we get 0.84. And the result seems to be close to the average result.
The snipers shot at the Rhino and failed to do damage. I was expecting this result after two successful turns in a row.
I think another article should go into assault averages because it is tricky. I won’t discuss the assault squad and tactical squad battle statistics now.
Ultramarine turn 3
Really no shooting done just finished up assault.
Dark Angles turn 4
Snipers failed to do damage again, as expected.
The rhino moved 12 inches so it couldn’t shoot.
Ultramarines turn 4
I shot at the snipers in the building because my opponent was feeling like I was winning by a bunch and so I gave him a 4+ roll to see who my speeders would shoot at and it ended up being the snipers. Some mercy in my playing can make the game feel more relaxed I think. Now onto the unsaved wound averages of this shooting attack. Krack missiles were used I think, any way frag missiles are hard to describe statistically when they scatter >1 inch, basically when they scatter off the base. Assuming I shot krak missiles. Also the scouts had a 3+ cover save in the building. Average unsaved wounds = (4/6)*(5/6)* (2/6)*2+(4/6)*(4/6)* (2/6)*6=1.26 . It looks like one of the scouts should have died on average.
Dark Angles turn 5
The snipers shot and did nothing to the scouts this time and did nothing. The scouts had cover saves 4+. Unsaved wounds on average =(4/6)*(3/6)*(3/6)*5=0.83. Hmm it looks like I should have lost a scout there.
The rhino rammed my rhino and killed it. It caused a strength 6 hit on the rhino. Average destroyed (there is nothing left on the rhino so all permanent damage will kill it) result = (1/6*2/6)+(1/6*4/6)=0.17. I think that the awesomeness of the ramming attack shifted the result away from the average in this case.
The tactical squad with the leader jumped out and shot up my land speeders doing quite a bit of damage, 2 shaken results and a weapon destroyed result. Now let’s look both the average result of both permanent damage and non permanent damage. Permanent damage average( Im going to lump the leader’s bolter shot in with the regular marines for ease of calculation, the actual result should be marginally higher) = (4/6)*((1/6)*(2/6))*9+(4/6)*((2/36)*(3/6)+(33/36)*(5/6))=0.86. And the result for non permanent damage is =(4/6)*((1/6)*(4/6))*9+(4/6)*((2/36)*(3/6)+(33/36)*(1/6))=0.79. So the weapon destroyed was to be expected, but the two shaken results are a bit above average.
Ultramarines turn 5
Mainly assaults happened this turn and that is a topic for a different day.
And that was the game guys. I hope this article was helpful in any way. If I made a numbers mistake in one of my calculations just comment on it and ill edit it out. If you have questions just ask em and I will do my best to answer them. I have attached some tables of some general statistics that are handy to know. The tables can be used to quickly find the: chance to hit, chance to damage,and chance to save. Which are the golden three numbers to find the average result.