# D&D 5EThe mathematics of D&D–Damage and HP

#### tetrasodium

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There are several other things to consider with attacks:
1. It is much easier to gain advantage on an attack roll than impose disadvantage on a saving throw.
2. There's no such thing as Legendary AC you have to burn through to score a hit.
3. Magic weapons are far more common than spell-boosting items.
4. There's no such thing as critical save damage
5. Multiple attacks benefit more from bonuses (whether magic items or spells like Enlarge)
6. Once you have a magic weapon, you never really need to worry about damage resistance again
So, on paper, it might not initially look like martials are that big a deal, but at the table (at least, didn't at a first reading of the 5e rules when I first got them), they really do dump out a lot of damage.
All good points & I'm glad that the thread didn't die after you posted that (very bad) chart back in 75 like it looked. You can't model damage over time with different hit rates using one line. The chart would need one line for each % chance to hit unless you are modeling something misleading or hard to visualize like damage over a specific number of rounds. I think it was probably a well meaning mistake rather than an attempt at obfuscation through "lies damn lies and statistics" though so moving on

The data below assumes +5 to relevant stat for both caster & martial. The specific damage type is irrelevant as it's easier to just model one with resistance & one without just as +1 +2 +1d6 weapons are easier to give their own lines. I'm going to give the martial the best odds to prove 5e still plunges into LFQW rather than inverting it though & say that the caster is using a d12 cantrip like toll the dead & the martial a d10 weapon like a longsword rather than modeling every possible choice.

I'm not going to make a pretty graph but I'll save others time & attach the sheet below
at 100% hit rate, right away you can see that the numbers aren't even close to supporting LFQW & at least with 100% hit rate/0% save chance it's massively inverted

..
Those numbers are massively skewed to an eye popping degree because every attack the longsword deals +5 from the relevant attrib and an extra +1 +2 or +3 while the cantrip never adds +5 from the relevant attrib and never adds the +1 +2 or +3 more than once per round

This next set is the same numbers with a 75% hit chance & 25% save chance. Since this is played out over rounds I don't need to model it by fiddling with average damage & can just show an accumulation of the numbers that actually deal damage. every third attack will miss & every fourth cantrip will save for none. Frankly the numbers at 100% are so bad that I questioned if there was even a point but decided to do it anyways for the sake of people who are struggling with the math (which is fine if someone is having trouble keeping up).

The first round of longsword data assumes there was a miss then every fourth attack or cast is a miss or save. That first round inverted for the cantrip with the first round assuming no save just because it was easier to model that way. Since this is showing damage over time in rounds it doesn't really matter.
The numbers are so stark that I'm not going to model what happens with only one attack with 75% hitrate & will jump straight to

at 50% the numbers are still horriffic & show just how much leveled spells need to bring to the table. If leveled spells were available in dramatically higher amounts that would be easy, but the numbers are so grossly out of wack that they need to be things like "every fireball hits the max number of targets and they mostly fail their saves" to bridge the gap because you can't cast a cantrip+a spell like you can attack+sneakattack or attack+action surge & already leveled spells barely just kinda break even rather than pull ahead

I tried to stack the deck in favor of showing LFQW was still a thing as much as possible & my reasoning was because for whatever reason Wotc chose to have both higher damage and more attacks on the weapon side of weapon to cantrip damage so the damage won't ever invert even if you get to very low hit rates. Paradoxically Wotc took that inverted LFQW one step further with the bewildering damage resistance magic resistance ac's almost guaranteed to hit excessive concentration use often unused by design spells & legendary resistance situation. If DR were still a thing on more than a handful of almost never used plant creatures or vulnerability to energy types were quite common there could conceivably be situations where the cantrip user would pull ahead but that's not the case either

edit:I originally flubbed the sheet by including strength looking at it & saying "oh I left out strength" to add it again

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#### fearsomepirate

##### Hero
All good points & I'm glad that the thread didn't die after you posted that (very bad) chart back in 75 like it looked. You can't model damage over time with different hit rates using one line. The chart would need one line for each % chance to hit unless you are modeling something misleading or hard to visualize like damage over a specific number of rounds. I think it was probably a well meaning mistake rather than an attempt at obfuscation through "lies damn lies and statistics" though so moving on

The chart was the Y% chance of doing at least X damage in a single round. I find this much more conceptually useful than DPR. Unfortunately, Anydice does not allow me to label axes. Probably should have been explicit about what they were.

So, for example, this chart shows that an 13th-level duelist fighter has a ~75% chance of doing at least 23 damage with his +2 longsword (target AC is 19 btw), while a wizard with Fire Bolt has a much lower chance , around 10%, of hitting that hard.

The reason I prefer this expression is that in a given fight, what is most important is whether you kill the enemy.

#### tetrasodium

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The chart was the Y% chance of doing at least X damage in a single round. I find this much more conceptually useful than DPR. Unfortunately, Anydice does not allow me to label axes. Probably should have been explicit about what they were.

So, for example, this chart shows that an 13th-level duelist fighter has a ~75% chance of doing at least 23 damage with his +2 longsword (target AC is 19 btw), while a wizard with Fire Bolt has a much lower chance , around 10%, of hitting that hard.

View attachment 134442

The reason I prefer this expression is that in a given fight, what is most important is whether you kill the enemy.
That explains why I was confused by your chart. I agree that damage per round is not an especially useful metric by itself as there are too many other variables to consider & that's why I did numbers for damage accumulated across rounds or what in WoW terms would be the total dataset of damage output over the course of a raid that gets used to calculate DPS, average damage, & so on.

WoW raids are a great example of why casters miss the mark so badly in 5e as it's organized by what everyone brings to the table then filling extra slots with quality of life stuff. in 5e we have the sword & board high hp high ac tank wearing both the crown for tank as well as having a damage output capable of laughing at a lower hp cloth wearing caster. That sort of situation is so flaming pants on head backwards & wrong that it runs straight into either "yea xyz creature is mostly immune to spells because X" or "we found a bug in the code & a htfix is rolling out shortly" type territory.

Extending the WoW metaphor it's fine for a class to have terrible damage output and terrible durability if it brings serious utility to the table such as battlefield control, buff/debuff, healing, or damage mitigation... but the way rests, the inability to drop below zero hp, & death by massive damage being pegged to max rather than current HP largely mitigates the last two there while spell slots & concentration along with monster ACs that are almost guaranteed to let a PC hit largely gets rid of the rest. Sure a well timed web can really save the day, but it can't be well timed if the baddies aren't all within a 20ft cube (smaller area than past editions when LFQW was still a thing) & the monsters get a save every round now and if they fail their save they just need to take 2d4 damage from one of the most common energy types or use their action to make a strength check.

One could say things like "but teleport" or some other noncombat utility spell & there might be a case to be made, but it's certainly not one supported by WotC's hardcovers when things like the wandering emporium don't even have a circle to learn.

#### fearsomepirate

##### Hero
The thing is that in 5e, combats are so few rounds that the law of large numbers doesn't apply to them. Sure, over the campaign, you'll do total damage in keeping with your DPR. But in a single encounter? You've got 3 to 5 turns, tops. So if the Wizard casts Banishment and fails, or a Rogue misses his attacks in the first and second rounds, everything can go sideways fast.

I am mostly pretty happy with casters 5e, as LFQW is dead, and martials now own the single-target damage niche. However, it can be hard for players to really grasp this, as damage dice simply have the lowest cognitive load. When faced with choosing between throwing a wad of damage dice on the table or various status effects, most players will simply choose damage, whether it's Magic Missile over Protection From Evil at low level, or Disintegrate over Otto's Irresistible Dance at high level.

#### tetrasodium

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The thing is that in 5e, combats are so few rounds that the law of large numbers doesn't apply to them. Sure, over the campaign, you'll do total damage in keeping with your DPR. But in a single encounter? You've got 3 to 5 turns, tops. So if the Wizard casts Banishment and fails, or a Rogue misses his attacks in the first and second rounds, everything can go sideways fast.

I am mostly pretty happy with casters 5e, as LFQW is dead, and martials now own the single-target damage niche. However, it can be hard for players to really grasp this, as damage dice simply have the lowest cognitive load. When faced with choosing between throwing a wad of damage dice on the table or various status effects, most players will simply choose damage, whether it's Magic Missile over Protection From Evil at low level, or Disintegrate over Otto's Irresistible Dance at high level.

Banish is on the spell list for cleric paladin warlock wizard and bonus spell lists for three paladin oaths so it's not like that's the wizard's saving grace or that other cantrip casters not named warlock don't have it just as bad as the wizard. That "if the wizard fails" is all the more reason having so many thumbs on the scale to make sure they never shine too bright even when it's something they should shine at is all the more problematic. Keep in mind that neither of our numbers included things like GWM & the hit rates for attacks targeting monster AC is so high that -5 to hit is still almost certain to hit in most cases. "If the wizard fails" they typically fail hard & they spend a spell slot to even try but if the caster does it's rarely all that impressive enough to bridge the gap in ways anyone cares about even if they succeed. Banishment is a symptom of the problem not a side road that avoids it

#### pemerton

##### Legend
@tetrasodium, in your charts where is the extra +5 damage coming from? Ie a longsword does 1d10 (weapon base) + 5 (stat) +n (magic) +5 (from what? critical? but why would the fighter be critting 100% of the time?)

I also had trouble following some of your formulas - eg in the 4 round column for 75% hit chance with two attacks, why does the damage step up by 32 (ie two round's worth) compared to the 3 round column? If this is Action Surge, why isn't it modelled in the 100% 1 attack column? From then on it seems that your columns step up in a +16, +32 pattern but why?

It seems to me that a fighter with +5 stat and +1 longsword with 95% chance to hit (best possible) does an average of 5.5 (base) +5 (stat) +1 (magic) + 5.5/19 (crit, which happens on a 20 ie 1 in 19 rolls) on a hit, = average of 11.8 (rounding up very slightly) on a hit = average of 11.2 per attack. What have I missed? - improved crit range will lift it a tiny bit; manoeuvre dice will lift it a noticeable bit; but I'm not seeing how you're lifting it to 16 per attack.

When you say your numbers show "how much leveled spells need to bring to the table" I think they ought to: if a wizard does the same damage as a fighter, or close to it, before those spells are brought to the table then the wizard will be obviously broken in their power, dominating in and out of combat!

#### fearsomepirate

##### Hero
Banish is on the spell list for cleric paladin warlock wizard and bonus spell lists for three paladin oaths so it's not like that's the wizard's saving grace or that other cantrip casters not named warlock don't have it just as bad as the wizard. That "if the wizard fails" is all the more reason having so many thumbs on the scale to make sure they never shine too bright even when it's something they should shine at is all the more problematic. Keep in mind that neither of our numbers included things like GWM & the hit rates for attacks targeting monster AC is so high that -5 to hit is still almost certain to hit in most cases. "If the wizard fails" they typically fail hard & they spend a spell slot to even try but if the caster does it's rarely all that impressive enough to bridge the gap in ways anyone cares about even if they succeed. Banishment is a symptom of the problem not a side road that avoids it

I didn't bring up Banishment as the be-all end-all of Wizard spells. I simply mentioned it as an example of why average-based analysis isn't all that helpful.

For the record, I haven't seen wizards struggle to shine in 5e. Outside of AoE, they do best when they think about how best to help the party rather than how best to roll a lot of damage dice.

#### tetrasodium

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@tetrasodium, in your charts where is the extra +5 damage coming from? Ie a longsword does 1d10 (weapon base) + 5 (stat) +n (magic) +5 (from what? critical? but why would the fighter be critting 100% of the time?)
Human error, I'll update the images in a minute

I also had trouble following some of your formulas - eg in the 4 round column for 75% hit chance with two attacks, why does the damage step up by 32 (ie two round's worth) compared to the 3 round column? If this is Action Surge, why isn't it modelled in the 100% 1 attack column? From then on it seems that your columns step up in a +16, +32 pattern but why?
Every third attack misses so there is one successful attack, two successful attacks one successful attack etc
It seems to me that a fighter with +5 stat and +1 longsword with 95% chance to hit (best possible) does an average of 5.5 (base) +5 (stat) +1 (magic) + 5.5/19 (crit, which happens on a 20 ie 1 in 19 rolls) on a hit, = average of 11.8 (rounding up very slightly) on a hit = average of 11.2 per attack. What have I missed? - improved crit range will lift it a tiny bit; manoeuvre dice will lift it a noticeable bit; but I'm not seeing how you're lifting it to 16 per attack.
I didn't include anything for crits because it complicates the math & would only skew even further
When you say your numbers show "how much leveled spells need to bring to the table" I think they ought to: if a wizard does the same damage as a fighter, or close to it, before those spells are brought to the table then the wizard will be obviously broken in their power, dominating in and out of combat!

#### tetrasodium

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I didn't bring up Banishment as the be-all end-all of Wizard spells. I simply mentioned it as an example of why average-based analysis isn't all that helpful.

For the record, I haven't seen wizards struggle to shine in 5e. Outside of AoE, they do best when they think about how best to help the party rather than how best to roll a lot of damage dice.
It's one of the better ways a wizard can do nondamage stuff, but casters have a huge damage gap significant hp gap & generally a notable AC gap so the nondamage stuff needs to bridge the gap & then some to make up for the times they are conserving spell slots or don't have the situationally perfect spell prepped.

Frankly there are too many thumbs on the scale for that to happen as combined they add up to an overwhelming combo
• Magic weapon of any kind ignores resist to nonmagical bps while energy resist is hugely overused but never goes away. This is bad enough that you almost immediately after Tasha's saw folks post order of scribes spell guides that list all the b/p/s spells & spells that go from rarely useful to great when they do things like switch from poison to bludgeoning damage.
• Concentration is overused on spells so for example web banish magic weapon & enlarge are all some level of ok to good spells but only one can be used as all of them are concentration
• Magic resist to give advantage against saves is overused while monsters with difficult to hit ACs are less than common.
• Feats exist to specialize for weapon damage (ie GWM & such) or defense (ie medium/heavy armor master) while very little specialization & synergy can be obtained through feats for casters with web+telekinetic get back in there one of the very few exceptions
• A huge number of spells generally fall under the "almost good" or "unused by design" labels, often due to excessive saves, secondary ways out, or range/aoe coverage that got shrunk to something that really kinda requires the GM to set the stage by putting all the baddies close together. Web is a good example of an ok to good spell that could have been great. Compare 3.5 & 5e web

• 10min per level to 1hr. This part is a wash since caster level 6 is nothing & will exceed it from 7th on but it's almost never going to be needed
• Reflex negates to dex negates is the same
• No saves each round to strength save each round
• DC20 strength check or DC25 escape artist check to dc=caster's spell save DC each round is a huge nerf that more than consumes the hit from move 5 feet for every 5 points you beat dc10 by since realistically caster dc is never going to be 20-25 in 5e.
• 20foot radius to a 20foot cube?... that's about half the aoe coverage
• 100ft+10ft/caster level to 60 ft?... That's a gigantic drop in range & especially notable given the insane range bump on ranged weapons.
• spell resistance:no to magic resist grants advantage on the save.... There don't even exist spells to bypass magic resistance in 5e, this is a gigantic nerf.
• fire & forget to concentration so your web might drop if you get his & even if that's not an issue you now can't cast web with most other buff/debuff/DoT/Area Denial spells like haste curse magic weapon elemental weapon enlarge wall of whatever(fire/ice/etc) & many more despite many of those spells themselves also getting a similar set of petty nerfs
• That concentration cherry on top of all the other petty nerfs & hinderances adds up to ensure the caster never uses them in conjunction to go from doing something to occasionally really stepping out from being the squishy character with mediocre damage into the spotlight as more than "well I guess that was a useful speedbump" or "I guess that helped a little"

#### pemerton

##### Legend
I don't have 5e play experience. I do have a lot of 4e play experience. In 4e, AoE action denial is very strong. The fact that it has a duration of only a round or two doesn't undermine its strength. I'm not aware of any reason to make me think that it would be markedly less strong in 5e.

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