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

#### fearsomepirate

##### Hero
Well, the point is that if you failed round 1, you wouldn't have a round 2 to change tactics on. You chose A because it was the attack that actually gave you the highest chance of surviving the encounter. If you have 2 rounds to live, you should choose attack B.

Not necessarily. If you can end things now, such as by casting Banishment on an extraplanar enemy, you save however many party resources would have been otherwise expended.

#### tetrasodium

##### Legend
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Epic
Immediate kills are more valuable than consistent kills for a particular reason: a dead enemy can't kill your teammates.

If you have an attack that guarantees a kill but it has a 50% chance to-hit (We'll call it attack A), you'd have a higher variance than an attack that guarantees a hit but takes 2 rounds to kill(which we'll call attack B). Which is better? Does it matter?

Yes, it does matter which one you choose. If we look at it with a sufficiently large number of trials, the percent misses to hits will equal about 50% and it will average to 2 rounds to kill on each, but we want an enemy dead as soon as it can be. With B, there's a 0% chance it will die round 1, a 100% chance it dies on round 2 and no more rounds have to be considered. But if you attack with A, you'd have a 50% chance to kill on round 1, a 25% chance to kill round 2, a 12.5% chance to kill round 3, etc.

What does this mean in terms of damage mitigated? Well, lets say the order is you first, monster second. The monster averages 100% of the damage round 1 and 0% damage round 2 for B. A has an average damage reduction of 50% for round 1 and 25% round 2, etc.

The significance is that depending on your own current survivability, you should choose one over the other.

If you can survive 2 rounds before resetting to full HP, you should do the second attack. But if you die as soon as the creature gets to attack, you should choose the first attack.

So now we can have our own survivability married to our decision of what is a good or bad attack. The average told us nothing, but the probabilities helped us understand the differences between A&B and which attack is best for which situation.

In general, if you're on your last ropes, you'll want to break out attacks with high variance. If you're certain that you won't be felled anytime soon, you'll want attacks with low variance.

And the application can now inform us that when your barbarian has high HP in a fight, he'll want to use his Greatsword. At low HP, however, he'll want to start swinging his Greataxe to increases the odds that he kills before he gets killed.
Depends. Take a 5th level caster with an 18 in the relevant casting stat casting fireball. It can do that twice & has a dc14 dex save compared to a fighter with a greatsword & 18 strength.
• A troll has 84 hp 15 ac & with +1 dex needs to roll a 14or better to save for half of an average 28hp. Both fireballs won't even bloody it. The fighter needs to roll an 8 or bettergiving him two attacks with a 60% chance of hitting for an average of 11 each. I started with troll because it's a good baseline example where both are pretty close but neither is obviously better without a lot of what if situational questions.
• There's no question that a fireball is great at smashing a bunch of clustered weak enemies so no need to do the math on what if an infinite number of 11hp goblins are clustered
• A barbed devil is the same ac15 cr5 as the troll but has 110 hp and due to magic resistance has advantage on a roll that needs to be 11 or higher for an effective 2 rolls each with a 55% chance of resulting in save for half or about a 60% chance to save for half while the fighter still has two swings each with a 60% chance of hitting but in this case the damage is halfed because the fighter doesn't yet have a magic greatsword
• By level 11 the caster has a good selection of spells that could be considered save or die & they both have a relevant 20, & a +1 greatsword/focus item respectively. Lets pit them against auril from icewind dale as it has a cr9 10 & 11 form

all three have legendary & magic resist
Nearly every save or die spell is going to be con or wis save, with those con/wis save bonuses on top of the resistances it's safe to call bothering with a save or die spell a wasted spell slot. With 13 16 & 19AC, the fighter has three attacks each with a +10 to attack where he needs to roll a 3 6 or 9 to hit, calling them almost certain almost certain & 55% chance of 3x13 slashing for the fighter is fair. Disintegrate & chain lightning both use a spell slot & if not saved one will slightly* outdamage the fighter for one round while the other if not saved will somewhat fall behind. Except the level 11 caster only has one sixth level spell so can't cast both of those in one fight yet.

5e really screws the pooch for the cost/benefit ratio math when it comes to one big hit vrs multiple small hits
*no seriously, that's not sarcasm. avg 45 & avg35+40 vrs GWM-5+10 +2d6+5*3= vg 55 assuming no crits needing a roll of 5 8 or 11 to hit. If the fighter crits then all bets are off & it doesn't really matter what the caster does

#### fearsomepirate

##### Hero
Sounds like the best use of a spell slot here is to cast Greater Invisibility on the Fighter. Next round, Etherealness on self and wish Casper the Friendly Ghost the best of luck.

#### tetrasodium

##### Legend
Supporter
Epic
Sounds like the best use of a spell slot here is to cast Greater Invisibility on the Fighter. Next round, Etherealness on self and wish Casper the Friendly Ghost the best of luck.
yea as much as 5e worries about the lfqw of past editions by setting up things like this, it almost needs very contrived situations to not stay inverted & even those contrived situations aren't all that inspiring if you do even basic ballpark eyeballing of the math. With the overuse of energy resists & near lack of energy vulnerabilities on common creatures beyond a certain CR it gets even worse. Auril is an unusual outlier with a vulnerability but there's nothing to suggest "oh obviously an ice spirit is vulnerable to radiant then fire then thunder." As much as wotc seems to eat up people with noncaster PCs complain about casters having so many options in combat compared to martials, the options are rarely anything but an illusion of choice or flashy spell slot consuming utterly average damage at best .

#### Asisreo

Not necessarily. If you can end things now, such as by casting Banishment on an extraplanar enemy, you save however many party resources would have been otherwise expended.
That's actually the same premise. If you can consistently do damage that will eventually equal or exceed the target's HP by 4 rounds but you can also cast Banishment with a 25% chance to hit, it may be more valuable to cast Banishment depending on reliability.

Now, if your chance to Banish was higher than 25%, then it may be safer to banish anyways, even if you have enough to survive 4+ rounds because you're more likely to save resources before it gets to that point.
Nearly every save or die spell is going to be con or wis save, with those con/wis save bonuses on top of the resistances it's safe to call bothering with a save or die spell a wasted spell slot. With 13 16 & 19AC, the fighter has three attacks each with a +10 to attack where he needs to roll a 3 6 or 9 to hit, calling them almost certain almost certain & 55% chance of 3x13 slashing for the fighter is fair. Disintegrate & chain lightning both use a spell slot & if not saved one will slightly* outdamage the fighter for one round while the other if not saved will somewhat fall behind. Except the level 11 caster only has one sixth level spell so can't cast both of those in one fight yet.
What you're referring to in the case of the Save or Die spells are what happens when the accuracy is different. When the accuracy of your random kill move is significantly lower than the accuracy of your consistent damage move, the damage move usually has priority.

Now, for the casting example, its true that AoE spells usually suck vs single enemies compared to single-target attack rolls from a typical melee combatant, but that's presumably on-purpose.

Well, first of all, we can actually determine the average number of creatures hit by an AoE using the DMG's formulae. For example, Fireball expects to hit (20/5)= 4 targets in range.

Now, I've actually been keeping an eye on how fireball compares to, say, a single-target character's attack. Actually fireball beats out a fighter's melee attack by alot, even accounting for GWM and extra attack...and I mean ALOT. This is assuming equal percent chance of success (I'll get to that).

Now, a fireball is more likely to kill any creature below 35HP, which is a pretty safe assumption against mooks. Above 35HP and the GWM fighter is more likely to kill.

But also, the average for Fireball is much, much higher than the average of GWM and it has less variance as well.

But what about unequal accuracies? Lets use the troll as an example:

+1 dex and 15 AC (84HP). Assuming level 6 and V.human , the fighter has a +8 to-hit. The wizard has a spell DC of 15, so the troll must roll a 14 or higher to save. In this particular instance, Fireball has the higher average damage, the lower variance, and has a higher kill chance for HP's below 17.

However, despite the higher average and other benefits of fireball, it turns out that above 17HP, GWM has a higher kill chance than fireball.

#### tetrasodium

##### Legend
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That's actually the same premise. If you can consistently do damage that will eventually equal or exceed the target's HP by 4 rounds but you can also cast Banishment with a 25% chance to hit, it may be more valuable to cast Banishment depending on reliability.

Now, if your chance to Banish was higher than 25%, then it may be safer to banish anyways, even if you have enough to survive 4+ rounds because you're more likely to save resources before it gets to that point.
What you're referring to in the case of the Save or Die spells are what happens when the accuracy is different. When the accuracy of your random kill move is significantly lower than the accuracy of your consistent damage move, the damage move usually has priority.

Now, for the casting example, its true that AoE spells usually suck vs single enemies compared to single-target attack rolls from a typical melee combatant, but that's presumably on-purpose.

Well, first of all, we can actually determine the average number of creatures hit by an AoE using the DMG's formulae. For example, Fireball expects to hit (20/5)= 4 targets in range.

Now, I've actually been keeping an eye on how fireball compares to, say, a single-target character's attack. Actually fireball beats out a fighter's melee attack by alot, even accounting for GWM and extra attack...and I mean ALOT. This is assuming equal percent chance of success (I'll get to that).

Now, a fireball is more likely to kill any creature below 35HP, which is a pretty safe assumption against mooks. Above 35HP and the GWM fighter is more likely to kill.

But also, the average for Fireball is much, much higher than the average of GWM and it has less variance as well.

But what about unequal accuracies? Lets use the troll as an example:

+1 dex and 15 AC (84HP). Assuming level 6 and V.human , the fighter has a +8 to-hit. The wizard has a spell DC of 15, so the troll must roll a 14 or higher to save. In this particular instance, Fireball has the higher average damage, the lower variance, and has a higher kill chance for HP's below 17.

However, despite the higher average and other benefits of fireball, it turns out that above 17HP, GWM has a higher kill chance than fireball.
All of that assumes that 4 or more targets will cluster up within a 20foot radius sphere and be creatures the party cares about. The fact that fireball is "intentionally overtuned". People use those AOE spells because they are ~2 levels higher than the spell slot

There are a handful of higher level spells I could include but it never really gets all that much better & the pattern continues with them so "use single target spells" is a bit hollow as most of those "intentionally overtuned" spells are aoe. For fun, think about where agonizing eldritch blast falls on that table at levels 5 11 & 17 too

#### Asisreo

All of that assumes that 4 or more targets will cluster up within a 20foot radius sphere and be creatures the party cares about.
Actually, I only did the case with single-target fireball. I realized that multiple creatures in the blast radius is still just a single-target case for each individual creatures. Total damage doesn't matter as much as individually targetted damage so you can just translate what a fireball would be like for four creatures by observing the individual cases.

The fact that fireball is "intentionally overtuned". People use those AOE spells because they are ~2 levels higher than the spell slot
View attachment 134046
The damage knob for fireball was definitely cranked a bit but there's actually no real spell of similar level that can compare to it by pure damage anyways. Those charts are purposefully undertuned, assumingly because WoTC doesn't want a DM to accidentally homebrew a spell that's basically "fireball 2.0" and blame the devs for giving guidance that completely breaks the game.

But don't think I'm disagreeing with your overall point. Caster's AoE's can be somewhat dependable when dealing with lower-level mooks but single-target damage dealers usually come out on-top against singular targets. I believe that's for the best in terms of balance.

Casters usually shine in either exploiting weaknesses or supporting the group, which is my favorite things to do in a TTRPG as a caster anyways.

There are a handful of higher level spells I could include but it never really gets all that much better & the pattern continues with them so "use single target spells" is a bit hollow as most of those "intentionally overtuned" spells are aoe. For fun, think about where agonizing eldritch blast falls on that table at levels 5 11 & 17 too
I feel agonizing blast gets focused on a bit too harshly considering it's costing a caster's cantrip, invocation slot, and action to use. A dedicated noncaster loses practically nothing going for AB equivalents and usually outdamages them anyways.

But it is still an incredibly powerful cantrip. Its just not a perfect spell.

#### tetrasodium

##### Legend
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Actually, I only did the case with single-target fireball. I realized that multiple creatures in the blast radius is still just a single-target case for each individual creatures. Total damage doesn't matter as much as individually targetted damage so you can just translate what a fireball would be like for four creatures by observing the individual cases.

The damage knob for fireball was definitely cranked a bit but there's actually no real spell of similar level that can compare to it by pure damage anyways. Those charts are purposefully undertuned, assumingly because WoTC doesn't want a DM to accidentally homebrew a spell that's basically "fireball 2.0" and blame the devs for giving guidance that completely breaks the game.

But don't think I'm disagreeing with your overall point. Caster's AoE's can be somewhat dependable when dealing with lower-level mooks but single-target damage dealers usually come out on-top against singular targets. I believe that's for the best in terms of balance.

Casters usually shine in either exploiting weaknesses or supporting the group, which is my favorite things to do in a TTRPG as a caster anyways.

I feel agonizing blast gets focused on a bit too harshly considering it's costing a caster's cantrip, invocation slot, and action to use. A dedicated noncaster loses practically nothing going for AB equivalents and usually outdamages them anyways.

But it is still an incredibly powerful cantrip. Its just not a perfect spell.
Yea the fact that those "iconic" spells are so over the top wipes out so many of what should be valid options. There's not really much in the way of "do I zap this one guy or those guys together" & the ace in my back pocket spells are things that are just stupid like heat metal. I used to enjoy the god wizard type exploiting weakness/supporting group type stuff in past editions but 5e is uniquely hostile to it with concentration & such. There's no more web these guys faerie fire that guy curse that other guy anymore when it really matters so it falls back to damage & things are a mess there.

shielding agonizing blast from well deserved scorn because it uses an invocation slot when you get 2 @3 3@5 4@7 5@10 6@12 7@ & 8@19 is a bit overly forgiving of something that should have never been a cantrip to scale with character level. At that point you might as well point out the weakness of needing to learn/scribe/prep fireball instead of some other spell

#### Asisreo

shielding agonizing blast from well deserved scorn because it uses an invocation slot when you get 2 @3 3@5 4@7 5@10 6@12 7@ & 8@19 is a bit overly forgiving of something that should have never been a cantrip to scale with character level. At that point you might as well point out the weakness of needing to learn/scribe/prep fireball instead of some other spell
I do actually consider the prep/known slot used up with spells like fireball. Its why playing a spellcaster can become widely inefficient especially when you're not well-suited for tactics and strategy.

I've disliked wizards in this edition as well but I recognize they can be useful especially for OoC stuff. Personally, I believe clerics are the best caster because they support much better than any other spellcaster with their mixture of great spells and features that can prevent the game from swinging too hard to recover.

#### tetrasodium

##### Legend
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I do actually consider the prep/known slot used up with spells like fireball. Its why playing a spellcaster can become widely inefficient especially when you're not well-suited for tactics and strategy.

I've disliked wizards in this edition as well but I recognize they can be useful especially for OoC stuff. Personally, I believe clerics are the best caster because they support much better than any other spellcaster with their mixture of great spells and features that can prevent the game from swinging too hard to recover.
I was comparing the invocation cost more to the very low cost of picking it for the spellbook to begin with more than preparing it & worded it badly trying to be inclusive for other casters. The shift from vancian to prepared really made a mess of things for the wizard though & wasn't always pretty for other casters yea . It's one of the things I've wanted to houserule back but wasn't sure about doing it given how ott so much of 5e is everywhere else. Kinda hoping that a5e at least incudes it as an option I wouldn't need to homebrew onto my now mostly newer players who can't just be told "like it used to be"

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