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So we had a discussion in another thread about which spell is better to focus on, but we got sidetracked by an argument about calculating the DPR of a spell.
(EDIT: This thread is about how to calculate spell damage into your DPR calculations, not about which spell is better. It's also a general equation, and will not include every nuance of spells - saves/resistances/etc.)
Let's assume we have a Hasted level 10 Magus casting Intensified Shocking Grasp:
We'll assume said Magus hits on a 7+ (70% chance) with their highest attack(s) in their routine and threatens a critical on a 15+ (30% chace).
So 3 attacks at 7+ - 70% chance to hit (Spell-Combat/Strike, 1st Iterative, Haste)
And 1 attack at 12+ - 45% chance to hit (2nd iterative).
The most obvious, and straightforward way is to calculate the chance to miss/hit/crit with the first attack, then calculate the chance to miss/hit/crit with the second attack (taking into account the chance that you'd already landed the spell and no longer have a spell to deliver with your attack), then continue this method until you run out of attacks. This is also easily the most time-consuming and difficult way to calculate your spell's DPR.
You'd end up with an equation like this:
30% chance to miss = 0 damage
49% chance to hit (no critical) = 35 damage
21% chance to hit (critical hit) = 70 damage
However if you miss the spell is still charged and continues on to the Second Attack:
30% chance to miss = 0 damage
49% chance to hit (no critical) = 35 damage
21% chance to hit (critical hit) = 70 damage
However these are the chances that you hit on the attack, not that you hit and the attack still has a spell charged, so the final numbers for this attack are only 30% of this:
9% chance to miss = 0 damage
14.7% chance to hit (no critical) = 35 damage
6.3% chance to hit (critical hit) = 70 damage
However if you miss the spell is still charged and continues on to the Third Attack:
30% chance to miss = 0 damage
49% chance to hit (no critical) = 35 damage
21% chance to hit (critical hit) = 70 damage
However these are the chances that you hit on the attack, not that you hit and the attack still has a spell charged, so the final numbers for this attack are only 30% of this:
2.7% chance to miss = 0 damage
4.41% chance to hit (no critical) = 35 damage
1.89% chance to hit (critical hit) = 70 damage
However if you miss the spell is still charged and continues on to the Fourth Attack. This attack has a lower chance to hit or to confirm a critical:
55% chance to miss = 0 damage
31.5% chance to hit (no critical) = 35 damage
13.5% chance to hit (critical hit) = 70 damage
However these are the chances that you hit on the attack, not that you hit and the attack still has a spell charged, so the final numbers for this attack are only 30% of this:
1.485% chance to miss = 0 damage
0.8505% chance to hit (no critical) = 35 damage
0.3645% chance to hit (critical hit) = 70 damage
So the numbers in bold give you the % chance to miss, to hit for regular damage (no critical), or to score critical hit. All we have multiply the chance to hit by the damage and then add them all together.
1.485% chance to miss = 0 damage
(49% + 14.7% + 4.41% + 0.8505%) chance to hit (no critical) = 35 damage
(21% + 6.3% + 1.89% + 0.3645%) chance to hit (critical hit) = 70 damage
(calculating)
1.485% = 0 damage
68.9605% = 35 damage
29.5545% = 70 damage
Now to get a DPR number from this we simply multiply the % chance by the corresponding damage, and add the totals together so:
(0.01485 X 0) + (0.689605 X 35) + (0.295545 X 70)
becomes
0 + 24.136175 + 20.68815 = 44.824325
Using this method we end up with 44.824325 DPR
Now that's not bad, we got there, and there's enough working out there that you should be able to check that I did everything right (and please do, I'm only human).
But who wants to do that much work? Who has the time to change the variables every time you level up, or change your buffs?
But there's an easier method:
First, Let's talk about confirming crits.
Because you don't lose the spell on a miss, the maths actually doesn't change much by changing your chance to hit.
If you only hit on a 15+ (30% chance to hit), then every hit it a critical threat. You then confirm on a 15+ (30%), so that means there's a 30% chance your Shocking Grasp is a crit. The 70% chance that you missed is irrelevant since you would just hold your spell for the next attack.
If you hit on a 9+ (60% chance to hit), then half your hits (50%) are critical threats. Of those critical threats 60% of them will confirm, so the equation is 50% × 60% = 30%. Therefore there is a 30% chance that any Shocking Grasp that lands will be a crit. The 40% chance that you missed is irrelevant since you would just hold your spell for the next attack.
If you hit on a 3+ (90% chance to hit), then one third of your hits (33.33%) are critical threats. Of those critical threats 90% of them will confirm, so the equation is 90% × 33.33% = 30%. Therefore there is a 30% chance that any Shocking Grasp that lands will be a crit. The 10% chance that you missed is irrelevant since you would just hold your spell for the next attack.
The only time this changes is when you need higher than a 15 to hit. If you only hit on an 18+ then every hit would be a critical threat, but you'd only confirm on an 18+ (15%), so you'd have a 15% chance to turn that Shocking Grasp into a crit.
So now that we know there's (almost) always a 30% chance that your spell becomes a critical hit, and a 70% chance that it does normal damage, we can put those numbers into the spell damage.
(0.7 X 35) + (0.3 X 70) = 45.5 damage.
What we want though is the simplest number here, so:
(0.7 X 35) + (0.3 X 70)
= (0.7 X 35) + (0.6 X 35)
= {(0.7 + 0.6} X 35}
= 1.3 X 35
This is the number we want.
Essentially any spell we cast and deliver with Spellstrike will deal (on average) 130% of the average damage of the spell.
Now we know the average damage for the spell, we need to know the average chance to hit. The average chance for the spell to hit is the same as the chance that AT LEAST ONE weapon attack hits, to find that we go:
So for our 10th level Magus with Haste, we have 4 attacks that hit (70%, 70%, 70%, 45%).
The chance that ALL of them miss is therefore:
30% X 30% X 30% X 55%
= 0.3 X 0.3 X 0.3 X 0.55
= 0.01485, or a 1.485% chance that all attacks miss (if you look in the first spoiler you'll see that this is the same number that was calculated there.
Now the chance that you HIT with the spell is:
100% - 1.485% = 98.515%
So now we have a chance to hit and a average damage of the spell, we can put that equation into action.
(Chance to hit) X (average damage) = DPR
(0.98515) X (45.5) = 44.824325
Using this method we end up with 44.824325 DPR
The perceptive among you will notice that this is the exactly the same number as using the other method.
This is a much easier method of calculating your DPR with Spell-Combat/Spellstrike. All you need is your chance to hit on all attacks (which you should have if you're doing DPR calculations) and your average damage for your spell (which you should have if you're doing DPR calculations).
You should be easily able to plug these into any DPR calculations, with any touch spell you care to use.
Final equation: (1-A) X (1.3 X B) = C
A = chance that all attacks miss
B = average spell damage
C = Spell-DPR
If you want to work out your total DPR it's just:
Weapon-DPR + Spell-DPR = Total-DPR.
Here are some examples:
Weapon DPR = 7.97555 (I'll let you find your own DPR calculators for that)
Spell DPR = (1-A) X (
A = 1 - (0.45 X 0.45) = 1 - 0.2025
B = 2d4 = 2 X 2.5 = 5
Therefore (1 - 0.2025) X (1.15 X 5) = 0.7975 X 5.75 = 4.585625
Spell DPR = 4.585625
Total DPR = Weapon DPR + Spell DPR
Total DPR = 7.97555 + 4.585625
Total DPR = 12.561175
Level 8 Magus with Shocking Grasp (not intensified) and no Haste. Hits on 8+, crits on 15+, deals 1d6+10 damage per hit.
Weapon DPR = 29.835
Spell DPR = (1-A) X (1.3 X B)
= (1 - 0.0735) X (1.3 X 17.5)
= 0.9265 X 22.75
Spell DPR = 21.077875
Total DPR = Weapon DPR + Spell DPR
= 29.835 + 21.077875
Total DPR = 50.912875
Level 15 Magus with Haste, using Close Range and Spell Perfection to deliver Maximized Disintegrate as a touch spell. Hits on a 4+, crits on a 15+, deals 1d6+22 on a hit.
Weapon DPR = 116.025
Spell DPR = (1-A) X (1.3 X B)
= (1 - 0.0008775) X (1.3 X 180)
= 0.9991225 X 234
Spell DPR = 233.794665
Total DPR = Weapon DPR + Spell DPR
= 116.025 + 233.794665
Total DPR = 349.819665
Obviously this doesn't take everything into account.
Shocking Grasp gives a +3 to hit vs metal armed/armoured opponents.
Corrosive Touch deals less damage vs enemies with Resistance to Acid.
Disintegrate deals significantly less damage if the target makes their fort save.
These are things you'd have to take into account for individual spells, but are beyond the scope of this equation.
Please feel free to check my maths by the way. One of the great things about mathematics is that there IS a right answer, so anyone with the know-how should be able to check the working. If there are any mistakes, or if anything is unclear let me know.
RPG Superstar 2009 Top 32
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This graph gives a nice comparison of Shocking Grasp damage compared to Frostbite.
Basically, when you're hasted then FB wins, except against enemies with very high AC (and against undead, obviously). HTH!
You need to adjust for holding a spell.
For example, you attack with a [intensified] shocking grasp and miss. The spell is still active and you can use an iterative attack to hit with it. Others give you multiple hits to discharge, and screw up the calculations. Lastly, do you apply a missed held spell to the next round's numbers? Much fun in the details.
/cevah
You need to adjust for holding a spell.
For example, you attack with a [intensified] shocking grasp and miss. The spell is still active and you can use an iterative attack to hit with it. Others give you multiple hits to discharge, and screw up the calculations.
Yup, that's taken into account.
(1-A) X (1.3 X B) = C
The "(1-A)" is the chance that the spell hits, including all Iterative, Haste, and Spell-Combat/Strike attacks.
(EDIT: It could also include any other attacks - such as AoO's - but you'd have to add them in. It should be all-inclusive.)
This was the crux of the argument in the other thread, and the reason I made this one. It's non-intuitive, but you don't need more detail than that short equation to give you 1 round's average Spell-DPR. It's also the reason I did damage calculations the long way at the top (so that you could see that we end up with the correct number). I'd have to brush up to turn it into an algebraic proof.
If you want to check it, try doing a long-form equation given any level and any (single touch) spell and see how it compares to the short equation. You should end up with the same number.
Lastly, do you apply a missed held spell to the next round's numbers? Much fun in the details.
/cevah
No this equation doesn't take that into account. By level 8 or so there should be less than ~2% chance that you miss every attack, so it shouldn't matter often. I also don't know that you'd really need that for a DPR calculator, since that's getting into really niche examples.
You could also still deliver the touch spell the next round before casting (by going "Combat-Spell" insdead of "Spell-Combat") and your chances of delivering the spell before you cast again go up to ~99.66% (assuming our Level 8-14 Magus with a 70% hit chance). So even in the worst case scenario this equation is still only ~0.34% inaccurate.
This was really just to share a much simpler equation to get your DPR calculations
I might look at it another day, but I'll need to be in the mood for it (no guarantees).
Finally, I don't know if I said it, but multi-touch spells (like Frostbite) don't need this equation, you simply add the spell-damage to the weapon damage for every attack and use a normal DPR calculator.
Elf, specifically for Elven Magic...
+2 to overcome Spell Resistance
Probably use Snowball with Close Range Arcana to ignore Spell Resistance in the first place, and Staggered is a nice touch.
Intensify Spell doesn't do you any good before you can get Close Range Arcana, so dedicate your cheater traits for Snowball.
VMC Sorcerer for the Elemental Bloodline, choose Earth or whatever it is that gives you Acid damage, if you get this ability via VMC...
Get the EWP Ioune Stone, use an Estoc with the Impact enchantment.
You are still somewhat nerfed against enemies resistant to Cold and Acid, but you still have Elven Magic giving you a +2 to overcome Spell Resistance with any/all your other spells.
Elf, specifically for Elven Magic...
+2 to overcome Spell ResistanceProbably use Snowball with Close Range Arcana to ignore Spell Resistance in the first place, and Staggered is a nice touch.
Intensify Spell doesn't do you any good before you can get Close Range Arcana, so dedicate your cheater traits for Snowball.
Ah, sure ... I'm not sure how relevant it is to this thread, but sounds good. I'd probably put resources elsewhere, since you can overcome Spell Resistance as a Magus just by hitting someone with a sword, but Snowball's not a bad spell (and hey I just discovered there are now 2 versions of Snowball).