How to calculate expected damage of an AoE effect

[Skrum]

I'm embarrassingly tripping on the math here -

Let's say I'm hitting a group of enemies with a fireball. The damage is 8d6 (average 28), and my save DC is 15. They have a +2 Dex save, giving them a 40% chance of hitting their save and reducing the damage by half. How do I calculate the total expected damage for N enemies (for simplicity, let's say N = 3).

[BeholderEyeDr]

Fully written out, you get:


TotalExpectedAvgDmg = #Targets x [ ( P(Save) x 0.5 x AvgDmg ) + ( P(Fail) x AvgDmg ) ]

So in your case, this is 3 x (0.4 x 0.5 x 28 + 0.6 x 28) = 67.2

[Skrum]

Awesome! I had a vague idea it could be calculated using an average (like 60/100 times it deals 28 damage, and 40/100 times it deals 14 damage, for a total of some hundreds of damage, then finally divided by 100 for the average). But this is way easier.

[Lvl 2 Expert]

There is a simpler formula for fireball damage that is nearly as accurate:

TotalExpectedAvgDmg = Lots

:smalltongue:

[LudicSavant]

Fully written out, you get:


TotalExpectedAvgDmg = #Targets x [ ( P(Save) x 0.5 x AvgDmg ) + ( P(Fail) x AvgDmg ) ]

So in your case, this is 3 x (0.4 x 0.5 x 28 + 0.6 x 28) = 67.2

This is close enough for practical intents and purposes.

To be fully 100% accurate, you would also need to account for the fact that 5e rounds down when halving odd damage rolls.

So, half of rolls (evens) get divided by 2, and half of rolls (odds) are divided by 2, then subtract 0.5. Conveniently, this averages out to -0.25 average damage (per target).

Anyways @Skrum, you can find the exact mathematical formula for this, as well as pretty much everything else in D&D, in the "Comprehensive DPR Calculator" post in my sig (written up by an actual mathematician). Said post can also be found stickied at the top of this forum (in "5e notable threads").

[Skrum]

Anyways @Skrum, you can find the exact mathematical formula for this, as well as pretty much everything else in D&D, in the "Comprehensive DPR Calculator" post in my sig (written up by an actual mathematician). Said post can also be found stickied at the top of this forum (in "5e notable threads").

Excellent. Tyvm.

In case anyone was interested why, I recently got a custom item that has the ability

Burn the Horde
1/LR, when casting a fire-based AoE spell, you may select a number of targets up to your prof bonus to have disadvantage on the save

This looks strong, but the more I thought about it the more it's actually not. So yeah, I wanted to do the math. Bringing in my estimates and character spell DC, it's worth about 10 damage. Once per LR. Rather weak, actually.

[stoutstien]

Excellent. Tyvm.

In case anyone was interested why, I recently got a custom item that has the ability

Burn the Horde
1/LR, when casting a fire-based AoE spell, you may select a number of targets up to your prof bonus to have disadvantage on the save

This looks strong, but the more I thought about it the more it's actually not. So yeah, I wanted to do the math. Bringing in my estimates and character spell DC, it's worth about 10 damage. Once per LR. Rather weak, actually.

I'd start backwards. Fine out roughly how much damage you want the features to provide and then find the formula that works.

[Skrum]

I'd start backwards. Fine out roughly how much damage you want the features to provide and then find the formula that works.

That would be a good thing to do - but I didn't make the item lol. My character received this item, and I was doing some analysis of it. Punchline, it's not worth the attunement slot unless I'm fighting undead (it has a few other abilities that work against undead).

[Tanarii]

Fully written out, you get:

TotalExpectedAvgDmg = #Targets x [ ( P(Save) x 0.5 x AvgDmg ) + ( P(Fail) x AvgDmg ) ]

P(Fail) = 1-P(Save)
TotalExpectedAvgDmg = #Targets x [ ( P(Save) x 0.5 x AvgDmg ) + ( {1 - P(Save)} x AvgDmg ) ]
TotalExpectedAvgDmg = #Targets x [ AvgDamage x (2 - P(Save) ) / 2]

P(Save) = (21+DexMod-DC) /20
TotalExpectedAvgDmg = #Targets x AvgDamage x (19 - DexMod + DC) / 40

[Kane0]

In case anyone was interested why, I recently got a custom item that has the ability

Burn the Horde
1/LR, when casting a fire-based AoE spell, you may select a number of targets up to your prof bonus to have disadvantage on the save


Seems fine, a bit specific even. The fire limitation part of the thematics?

[Amechra]

I mean, a +15% damage bonus during an important fight is nothing to sneeze at!

The more important question: how many other magic items do you have that need attunement? Looking at the numbers, you're [I]probably 5th/6th level — I'd be surprised if your attunement slots are already full.

[Skrum]

I mean, a +15% damage bonus during an important fight is nothing to sneeze at!

The more important question: how many other magic items do you have that need attunement? Looking at the numbers, you're [I]probably 5th/6th level — I'd be surprised if your attunement slots are already full.

I'm level 8, and already have 3.

This game has a crafting system wherein we can spend session XP on making (approved) published items, and then custom items can also be found at the various DM's discretion (there's more than one DM).

The Gauntlet
+1 universal casting focus (+1 to hit and DC's)
1/LR can cast protection from good and evil

PB/LR when rolling damage for a fire spell you may add your casting stat to the damage against undead foes

1/LR (burn the horde)

So, I have Rod of the Pact Keeper, a ring of protection, and a custom sword for my attunement slots. My character is a fighter/warlock gish that uses a lot of fireballs. This item was obviously made with him in mind.

The sword is by far my best item, so that's not going away. The ring and rod is boring, but I think probably better than the gauntlet unless I know I'm facing undead. If I know I am, I'll probably drop the rod and switch to the gauntlet.

[Tanarii]

The Gauntlet
+1 universal casting focus (+1 to hit and DC's)


PB/LR when rolling damage for a fire spell you may add your casting stat to the damage against undead foes

The first is worth = #Targets x AvgDamage / 40

Second is worth = CastingStat x #Targets x (19 - DexMod + DC) / 40

Disadvantage on the save changes things enough it's worth just sticking with the original P(save) formula, and subtracting the normal result from the disadvantaged result for the number of affected disadvantaged targets:
TotalExpectedAvgDmg = #Targets x [ AvgDamage x (2 - P(Save) ) / 2]

[Skrum]

I think I've actually changed my mind on this item - I'm gonna drop the Rod and use the gauntlets full time. My character uses a shield and sword, so the rod is a PITA. The gauntlets give +1 to my spells without taking a hand slot, plus its other abilities. Losing the effective extra spell slot hurts, but at least ill get a free cast of PfGE (which I use fairly often).