Guest writer William Ford takes a close mathematical look at the Divination psychic power for Warhammer 40,000 and how being smart and junk can help you write those army lists more efficiently.
Conventional internet wisdom says that Divination is a gift from the emperor himself. Let’s look at some numbers and figure out when Divination is going to be most useful. Not all units are created equal, so not everyone is going to get the same benefit. The two powers I am concerned with in this problem are prescience and forewarning. They offer the most stable versions of boosting damage and survivability.
With armor saves I am going to assume:
6+ armor gets ignored 95% of the time
5+ armor gets ignored 80% of the time
3+ armor gets ignored 25% of the time
The Biggest Assumption
All of the percents shown here are a single squad being compared with itself. None of the percents here are intended to be compared with other squads. A unit might have their damage increased by 50%, but if all they have are lasguns then how does this compare to a unit of bolters that increases their damage by 25%? That is a topic for another day.
The Math Situation **Avoid if math scares you**
If a power increases durability by 30% and has a 50% to be cast then we multiply 0.3*0.5 and get an overall improvement of 0.15 or 15%. Now let’s say that because of our psychic powers we improve a squad’s durability by 15% and damage output by 40% we would multiply 1.15*1.40=1.61. This means that if we increase a squads damage output by 40% and the durability by 15% then we have actually increased its overall performance by 61%. The reason we multiply rather than add 0.15+0.40=0.55 is because if a squad is more durable, then they are going to survive longer to shoot more.
The chance to cast powers is also dependent on the chances of getting Forewarning. For instance when I say that a level 1 psyker has a 66.19% chance to cast prescience and a 6.94% chance to cast Forewarning this is averaging several things. First off I am accounting for random amounts of warp charges, and I am averaging the chances for all of these charges. Second off I am also accounting for when Forewarning is successfully rolled as well as when it is not. If we have Forewarning, then the number of dice we use on Prescience is going to be less than before.
Primaris Work:
A level 1 Primaris Psyker will have an overall chance to cast prescience of 66.19% and a 6.94% chance of getting Forewarning, having enough warp charges, and successfully casting it. When the Primaris is upgraded to level 2 these chances are increased to 74.48% and 19.1% respectively.
Librarian Work:
The chances to cast powers for librarians are the same as they are for a Primaris Psyker.
Sorcerer work:
In order for a sorcerer to get Divination it needs to take the Balestar of Mannon, which has the benefit of re-rolling failed psychic checks. A level 1 Sorcerer will cast Prescience 82.19% of the time and forewarning 11.46% of the time. Similarly a level two Sorcerer will cast prescience and forewarning 89.06% and 28.45% of the time respectively. Lastly a level three Sorcerer will cast Prescience 94.35% of the time and forewarning 44.73% of the time.
To go from ‘overall benefit’ to ‘minimum squad cost’ I take the base cost of a psyker and divide by the overall benefit (written as a decimal). For instance (Cost of a level one Primaris Psyker)/0.5312=94.12. This means that a squad of guardsmen that costs more than 94 points is going to benefit from having a Psyker.
**Math concluded (mostly)**
Blob Squads with Primaris Psykers
| Primaris Psyker | Offense | Defense | Overall | Minimum Squad Cost |
| LvL 1 | 33.10% | 6.25% | 41.42% | 121 pts |
| LvL 2 | 37.24% | 17.19% | 60.83% | 123 pts |
Prescience will increase a guardsman’s chance to hit by 50%, and when this is combined with the previously mentioned chances to cast we get the offense bonus shown here. Forewarning will increase defense by 100% for any shots that ignore the 5+ save and increase defense by 50% for all other shots.
Marines with a Librarian
| Librarian | Offense | Defense | Overall | Minimum Squad Cost |
| LvL 1 | 22.06% | 1.74% | 24.18% | 268 pts |
| LvL 2 | 24.83% | 4.78% | 30.80% | 292 pts |
With Marines’ better ballistic skill, prescience is only going to increase their chance to hit by 33%. Once again forewarning is going to provide 100% increased defense when our armor is ignored, but isn’t going to provide any protection against normal damage.
Since marines already have a save and decent ballistic skill they aren’t going to receive as much benefit as the guardsmen did, and as a whole librarians are not a good combination for marine equivalents. Their cost is too great in comparison to the overall performance boost. In general it seems that buying additional squads is going to give you better results.
Chaos Marines and Sorcerers
| Sorcerer | Offense | Defense | Overall | Minimum Squad Cost |
| LvL 1 | 27.39% | 2.87% | 31.05% | 289 pts |
| LvL 2 | 29.73% | 7.11% | 38.95% | 295 pts |
| LvL 3 | 31.45% | 11.18% | 46.15% | 303 pts |
Chaos marines should theoretically see the same benefit as their loyalist counterparts, but the increased cost of a Balestar of Mannon increases the minimum squad cost and increases the chances of casting powers.
Renegade Legions and Sorcerers
Renegade legions are an interesting situation because they have the option to improve both their ballistic skill and their armor save, so I will have one set of info for a unit with no upgrades and one with +1BS and a 6+ armor save.
| Sorcerer | Offense | Defense | Overall | Minimum Squad Cost |
| LvL 1 | 54.79% | 11.46% | 72.53% | 124 pts |
| LvL 2 | 59.37% | 28.45% | 105.00% | 110 pts |
| LvL 3 | 62.89% | 44.73% | 136.00% | 103 pts |
An un-upgraded squad of renegades are going to increase their offensive power by 66.66% with prescience and their defensive power by 100% from forewarning.
Despite the increased cost of additional psychic levels the benefit is so significant that the total cost of the recommended squad goes down!
| Sorcerer | Offense | Defense | Overall | Minimum Squad Cost |
| LvL 1 | 41.10% | 11.27% | 57.00% | 158 pts |
| LvL 2 | 44.53% | 27.97% | 84.00% | 137 pts |
| LvL 3 | 47.18% | 43.98% | 111.00% | 126 pts |
An upgraded squad sees the same effect, but it isn’t as dramatic. Since the squad has an improved ballistic skill and an armor save they don’t receive quite as much benefit from the psychic powers. Prescience will only improve their chance to hit by 50% and forewarning will improve their chance to save by 98.33%
Remember, even though this situation appears to be very similar to that of the un-upgraded squad of renegades we can’t effectively compare the two squads with the information here.
What we can say is that most of the time it isn’t worth the points to buy a squad a renegades a Sorcerer. The minimum squad cost is significantly lower than what we saw with the marines, but renegades are so cheap that Sorcerers are simply too expensive.
Leman Russ Squads
| Primaris Psyker | Offense | Defense | Overall | Minimum Squad Cost |
| LvL 1 | 38.60% | 6.94% | 48.20% | 103 pts |
| LvL 2 | 44.61% | 19.10% | 72.23% | 103 pts |
| Sorcerer | Offense | Defense | Overall | Minimum Squad Cost |
| LvL 1 | 52.50% | 11.46% | 69.97% | 128 pts |
| LvL 2 | 56.89% | 28.45% | 101.52% | 113 pts |
| LvL 3 | 60.27% | 44.73% | 131.96% | 106 pts |
What we see here is that regardless of whether you have loyalist or traitor tanks; having a Psyker dedicated to buffing your squad is a good idea. These numbers were run based on trying to hit with a blast weapon. Since scatter is random I can’t say that that you are going to increase damage by x%, but I can say that you will increase your chance to hit by x%. Weapons that hit with BS will gain slightly less from prescience, but the defensive benefit will be the same. Either way buffing your Leman Russ squads is beneficial.
Artillery
| Primaris Psyker | Offense | Defense | Overall | Minimum Squad Cost |
| LvL 1 | 44.12% | 6.94% | 54.12% | 92 pts |
| LvL 2 | 50.98% | 19.10% | 90.30% | 83 pts |
| Sorcerer | Offense | Defense | Overall | Minimum Squad Cost |
| LvL 1 | 54.79% | 11.46% | 72.53% | 124 pts |
| LvL 2 | 59.37% | 28.45% | 105.00% | 110 pts |
| LvL 3 | 62.89% | 44.73% | 136.00% | 103 pts |
Just like the previous situation we are looking at increased chance to hit for the ‘offense’ column. Once again the benefits are significant enough where taking a dedicated Psyker to buff an artillery battery is a good idea. Obviously units such as Wyverns that already have ‘twin linked’ aren’t going to benefit from Prescience.
Summary
In a practical list building standpoint you are going to be limited on both points and models, but here is how we can connect the abstract to something a bit more practical.
First off, when a psyker is worthwhile it is also generally worthwhile to max out their level. Many squads had the minimum cost increase only by a slight amount, or in some cases it actually decreased. In a world of min/maxing it is rare that the middle ground is the ideal choice, and looking at many lists, you don’t see a lot of partially upgraded psykers.
The second point is a bit more obvious. Units that have low ballistic skill and low saves are going to benefit more than units with higher stats. If you have two units with a lascannon you are going gain more benefit buffing the unit with lower ballistic skill. The same idea follows with armor saves. Units with worse armor saves have more to gain from forewarning than units with better armor saves. As a result, vehicles are great targets for forewarning.
This is all subject to change when you consider what units are going to be attacked. If a unit of space marines is going to charge some terminators, then it is probably good to cast forewarning on them over a unit of renegades who aren’t going to get attacked. Also in an actual game you don’t need to cast both powers on the same unit. You can min/max whatever units need buffs at that particular moment.
Let us know in the comments section if you use similar math to make effective army lists in Warhammer 40,000.
>With armor saves I am going to assume:
>6+ armor gets ignored 95% of the time
>5+ armor gets ignored 80% of the time
>3+ armor gets ignored 25% of the time
This is a massive assumption to make without any kind of justification, and since it underlies all the math that supposedly drives all of the other calculations it seems like a rather problematic one to make.
It is an absolutely massive assumption, but none of the other calculations for forewarning are possible without some kind of assumption of this kind.
If your model doesn’t work without an invalid assumption, perhaps that indicates your model is flawed.
That’s not true at all. You can still make calculations like:
“If the save isn’t ignored, your defense is increased by x%”
“If the save is ignored, your defense is instead increased by y%”
But using the assumptions to say “I predict your defense, on average, is increased by this much” is a bit disingenous if there isn’t a solid basis for the original assumptions.
I agree armor-ignoring numbers are a huge assumption, and I feel there’s a lot more AP2 in competitive play (thank you Riptides et al). The good news is this assumption affects only the Forewarning numbers, not Prescience.
But intuitively I agree with the Conclusions at the bottom. Maxing psyker levels is conventional wisdom, and casting Prescience on units with a high firepower to Ballistic Skill ratio should be common sense.
The article attempts adding more value in the break-even point values, and those heavily rely upon that first assumption. If we assumed 40% AP2, which way would those point values move? Lower, justifying more psykers fishing for Forewarning?
I’m not sure this math realy leads to the right conclusion. Yes you do get the most benefit out of buffing up a weaker unit, in theory.
Yet in practice we often end up shooting for things like 2+ rerollables. Because at that pont the failure rate drops to more or less zero which means the rolls are reliable. A theoretical boost that will vary over games is good but in a five round setting being reliable is everything. The unit doesn’t just have to be good. It has to do its job all the time, and preferably never fail (nothing ever gets there I know).
Yeah, this is always something to consider. Sure, buffing my BS3 squad of guardsmen with TL is going to lead to the biggest % increase in hits. But I really don’t care if I have double the hit rate with lasguns. I really care about that devastator squad that only gets 4 lascannon shots and really needs to make them count. If I get unlucky and 2 or 3 miss, I’m hosed. I need them to almost all hit in order for the squad to do anything.
My “2.3 expected hullpoints” doesn’t mean crap if in practice it means one round of shooting I get 0 and the vehicle doesn’t die.
This all seems logical to me. Rerolling is better, if you don’t need the reroll then it isn’t as useful. Seeing some numbers on how much under a baseline environment is interesting. Maybe 40k is so realistic (gasp!) that not every possibility can be accounted for in a two page article.
Definatly not. But I think there is the problem? It feels like the article was reaching a bit high in what is possible to pin down with numbers like that.
My brain hurts