D&D 5E The mathematics of D&D–Damage and HP

[The Grassy Gnoll]

This is the rough framework I use if it’s any help or use
Meat
A character’s ‘meat’ hit points is always the same: starting HP (inc CON mod). The rest is skill, luck, etc.

• Any damage taken below that level is wound damage.
• Any damage of that amount or more is described as a wound.
• Thus, narratively, at L1 and even L2, most if not every hit taken will be narrated as a wound of some sort. As you might expect for a low level PC.
• Their ‘ability’ to avoid such wounds increases with level advancement.

At L1:
L1 Fighter (Fred) assuming 14 Con = 12HP
L1 Sorcerer (Sue) assuming 12 Con = 7HP
Any hit they take is narrated as actual damage. L1 is scary!

Beyond L1:
Fred (28 max HP) and Sue (17 max HP) are now L3. Fighting a hobgoblin guard, Fred is hit four times in the course of the battle, and Sue just the once (the hobgoblin was focused on the hairy bastard with the big sword rather than the civilian-looking one).

• Hit 1 on Fred: 6 dmg (HP now 22)
• Hit 2 on Fred: 5 dmg (HP now 17)
• Hit 3 on Fred: 4 dmg (HP now 13)
• Hit 4 on Fred: 2 dmg (HP now 11 - below his ‘meat’ of 12HP)

Hits 1-3 narrated as bruising, knocked breathless, etc. Hit 4 narrated as e.g., a (in this case, as it’s only 2HP, weak) cut to the forearm.

• Hit 1 on Sue, however: 7 dmg (HP now 10).

As this is her ‘meat’ amount, it might be narrated as a nasty gash to the shoulder.

Adding more meat
To reflect a more experienced PC’s ability to avoid getting those cuts etc as they gain levels, you can tinker a bit.
Eg

• If a PC takes a CON ASI, you might consider adding an extra amount to their ‘meat’ points - eg at L4, Sue chooses to boost her CON to 14, so her ‘meat’ increases from 7 to 8.
• Every time a PC reaches a Prof bonus watershed, add 1 to their ‘meat’ points
• This does, conversely, mean that high level PCs will suffer wounds at a higher level of remaining HP than a low level character.
• Eg Sue at L2 is wounded when she’s reduced to 7 HP; at L4 having taken the above ASI, she’s wounded ‘earlier’ at 8 HP.

NPCs and Monsters
Same as above really but base on Hit Dice.

• ‘Meat’: max HD die + CON bonus + CR if 1+

Eg
Tarrasque HP 676 (33d20+330), CON bonus of +10
CR 30
‘Meat’ HP therefore (20+10+30) = 60 HP
Wound narrated when takes 60+dmg in 1 attack or reduced to 60HP or fewer (roughly 1/11 max HP).

Young green dragon HP 136 (16d10+48), CON +3
CR 8
‘Meat’ HP therefore (10+3+8)= 21 HP
Wound narrated when takes 21+dmg in 1 attack or reduced to 21HP or fewer (roughly 1/7 max HP).

Bulette HP 95 (9d10+45) with Con bonus of +5
CR 5
‘Meat’ HP therefore (10+5+5) = 20 HP
Wound narrated when takes 20+dmg in 1 attack or reduced to 20HP or fewer (roughly 1/5 max HP).

Manticore HP 68 (8d10+24), CON bonus of +3
CR 3
‘Meat’ HP therefore (10+3+3) = 16 HP
Wound narrated when takes 16+dmg in 1 attack or reduced to 16HP or fewer (roughly 1/4 max HP).

Owlbear HP 59 (7d10+21), CON bonus of +3
CR 3
‘Meat’ HP therefore (10+3+3) = 16 HP
Wound narrated when takes 16+dmg in 1 attack or reduced to 16HP or fewer (just under 1/4 max HP).

Bugbear HP 27 (5d8+5) with Con bonus of +1
CR 1
‘Meat’ HP therefore (8+1+1) = 10 HP
Wound narrated when takes 10+dmg in 1 attack or reduced to 10HP or fewer (roughly 1/3 max HP).

Orc HP 15 (2d8+6), CON bonus of +3
CR 1/2
‘Meat’ HP therefore (8+3+0) = 11 HP
Wound narrated when takes 11+dmg in 1 attack or reduced to 11HP or fewer (roughly 2/3 max HP).

Goblin HP 7 (2d6), CON bonus +0
CR 1/4
‘Meat’ HP therefore (6+0+0)= 6 HP
Wound narrated when takes 6+dmg in 1 attack or reduced to 6HP or fewer (basically any damage taken).

No, not very scientific but as a rule of thumb it works for me.

 

[fearsomepirate]

Here's how I narrate them:

Player: Two attacks, a 12 and an 18.
Me: One hit. Damage?
Player: 14
Me: OK. Anything else?
Player: Nope
Me: OK, rogue's turn.

Riveting stuff, I know.

 

[Asisreo]

Alright, here we go.

The Math of Extra Attack.

So speaking of averages, standard deviations, and probabilities; I feel we should discuss the distinction between Extra Attack and High Damage single-attacks.

First, Let's take a look at a Fighter with a duel-wielding shortsword versus a Fighter with a two-handed greatsword. Assuming the same to-hit and no fighting style, they should be equal to each other, right? Again, yes but only in terms of average damage.

Let's say you're fighting a monster with 15 AC and you have a +5 to-hit with a +3 damage modifier. The monster has 28 HP. Which one is better?

Remember my example about consistent damage versus swingier damage? Well, the Greatsword not only has a higher variance, but the chances to hit with the greatsword is much lower. Greatsword has a 45% chance of completely missing while TWF has a 20.25% chance of missing. You're more than doubling your chances to-hit. A more guaranteed damage reduction (when a monster isn't at kill range or your character isn't close to dying) is more valuable than swingier high damage because you reduce the chances of unnecessary additional rounds from the enemy while also more predictably reducing them into the kill range for follow-up attacks.

Two-weapon fighting obviously has the cost of a bonus action to use, but if your character has no need for their Bonus Action, its a nice boost in damage. Remember that characters like Barbarians can dual-wield whenever they'd like. Of course, there's also turns where even if you do have a bonus action use, its not as good as just getting extra damage. Like a rogue that ran 30ft to do a dagger attack might as well use their BA to attack with their other hand since they can't make use of disengage or hide.

But also, I want to discuss how attacking twice for (1d6+3)+1d6 is NOT the same as 2d6+3 because of how accuracy works. Why?

When you sum two independent random variable's probability distributions, you take their convolutions. In simpler, but somewhat inaccurate terms: its the process that turns 1d6 to 4d6 into a bell curve rather than a larger rectangle. Now, the reason why this changes the probability curves isn't because of the damage dice, its the accuracy and the Order of Operations.

For the case of extra attack, you first take the probability distribution adjusted by the accuracy, then you convolute. In the case of the greatsword, you first convolute the probability distribution not adjusted by the accuracy, then you account for accuracy.

The result is that both have the same averages (over a large number of trials, the averages converge), but the Greatsword has a much higher variance and therefore more suited to killing enemies during last-ditch efforts and dual-wielding is more suited for consistently damaging enemies in an attrition.

 

[tetrasodium]

Alright, here we go.

The Math of Extra Attack.

So speaking of averages, standard deviations, and probabilities; I feel we should discuss the distinction between Extra Attack and High Damage single-attacks.

First, Let's take a look at a Fighter with a duel-wielding shortsword versus a Fighter with a two-handed greatsword. Assuming the same to-hit and no fighting style, they should be equal to each other, right? Again, yes but only in terms of average damage.

Let's say you're fighting a monster with 15 AC and you have a +5 to-hit with a +3 damage modifier. The monster has 28 HP. Which one is better?

Remember my example about consistent damage versus swingier damage? Well, the Greatsword not only has a higher variance, but the chances to hit with the greatsword is much lower. Greatsword has a 45% chance of completely missing while TWF has a 20.25% chance of missing. You're more than doubling your chances to-hit. A more guaranteed damage reduction (when a monster isn't at kill range or your character isn't close to dying) is more valuable than swingier high damage because you reduce the chances of unnecessary additional rounds from the enemy while also more predictably reducing them into the kill range for follow-up attacks.

Two-weapon fighting obviously has the cost of a bonus action to use, but if your character has no need for their Bonus Action, its a nice boost in damage. Remember that characters like Barbarians can dual-wield whenever they'd like. Of course, there's also turns where even if you do have a bonus action use, its not as good as just getting extra damage. Like a rogue that ran 30ft to do a dagger attack might as well use their BA to attack with their other hand since they can't make use of disengage or hide.

But also, I want to discuss how attacking twice for (1d6+3)+1d6 is NOT the same as 2d6+3 because of how accuracy works. Why?

When you sum two independent random variable's probability distributions, you take their convolutions. In simpler, but somewhat inaccurate terms: its the process that turns 1d6 to 4d6 into a bell curve rather than a larger rectangle. Now, the reason why this changes the probability curves isn't because of the damage dice, its the accuracy and the Order of Operations.

For the case of extra attack, you first take the probability distribution adjusted by the accuracy, then you convolute. In the case of the greatsword, you first convolute the probability distribution not adjusted by the accuracy, then you account for accuracy.

The result is that both have the same averages (over a large number of trials, the averages converge), but the Greatsword has a much higher variance and therefore more suited to killing enemies during last-ditch efforts and dual-wielding is more suited for consistently damaging enemies in an attrition.

In the past this used to be balanced out with dr/x & resist/x since hitting something repeatedly in a turn for smaller chunks is going to eat it more often than fewer all or nothing attacks. 5e changed that so they both suffer half damage to each while the chances to hit for a fraction are still a stregth/weakness jut the same.

 

[fearsomepirate]

Alright, here we go.

The Math of Extra Attack.

So speaking of averages, standard deviations, and probabilities; I feel we should discuss the distinction between Extra Attack and High Damage single-attacks.

First, Let's take a look at a Fighter with a duel-wielding shortsword versus a Fighter with a two-handed greatsword. Assuming the same to-hit and no fighting style, they should be equal to each other, right? Again, yes but only in terms of average damage.

Let's say you're fighting a monster with 15 AC and you have a +5 to-hit with a +3 damage modifier. The monster has 28 HP. Which one is better?

Remember my example about consistent damage versus swingier damage? Well, the Greatsword not only has a higher variance, but the chances to hit with the greatsword is much lower. Greatsword has a 45% chance of completely missing while TWF has a 20.25% chance of missing. You're more than doubling your chances to-hit. A more guaranteed damage reduction (when a monster isn't at kill range or your character isn't close to dying) is more valuable than swingier high damage because you reduce the chances of unnecessary additional rounds from the enemy while also more predictably reducing them into the kill range for follow-up attacks.

Two-weapon fighting obviously has the cost of a bonus action to use, but if your character has no need for their Bonus Action, its a nice boost in damage. Remember that characters like Barbarians can dual-wield whenever they'd like. Of course, there's also turns where even if you do have a bonus action use, its not as good as just getting extra damage. Like a rogue that ran 30ft to do a dagger attack might as well use their BA to attack with their other hand since they can't make use of disengage or hide.

But also, I want to discuss how attacking twice for (1d6+3)+1d6 is NOT the same as 2d6+3 because of how accuracy works. Why?

When you sum two independent random variable's probability distributions, you take their convolutions. In simpler, but somewhat inaccurate terms: its the process that turns 1d6 to 4d6 into a bell curve rather than a larger rectangle. Now, the reason why this changes the probability curves isn't because of the damage dice, its the accuracy and the Order of Operations.

For the case of extra attack, you first take the probability distribution adjusted by the accuracy, then you convolute. In the case of the greatsword, you first convolute the probability distribution not adjusted by the accuracy, then you account for accuracy.

The result is that both have the same averages (over a large number of trials, the averages converge), but the Greatsword has a much higher variance and therefore more suited to killing enemies during last-ditch efforts and dual-wielding is more suited for consistently damaging enemies in an attrition.

Picture's worth a thousand words:

Note also that multiple attacks are especially important when you have extra damage abilities that trigger on a hit.

 

[tetrasodium]

Picture's worth a thousand words:
View attachment 134162

Note also that multiple attacks are especially important when you have extra damage abilities that trigger on a hit.

Weapons have one by default while csntrips do not. +weapons and class abilities add on top of that while resist non magic bps quickly goes away &magic resistance/legendary resist /energy resistances always apply

 

[Asisreo]

Now two standard deviations for a normal distribution include 94% of the distribution, or about your odds of not rolling a 1 (or a 20).

I am willing to nitpick your nitpick and actually discuss further:

Neither of the damage distributions are a "normal" distribution because of both accuracy, criticals, and the effects hits and misses have on damage. Because they aren't bell-curved, the Empirical Rule for standard deviation doesn't apply.

We can't say that 94% of data lies within 2 standard deviations but we can use Chebyshev's Thereom to calculate what percentage of data is at least within a certain number of standard deviations.

For 1 SD, we get at least 0% of the data lies within a single SD. For 2 SD, at least 75% of data lies within 2 standard deviations. For 3 SD, at least 88.89% of data lies within 3 standard deviations.

Just a small mathematical nitpick.

 

[el-remmen]

I always roll for damage and love so-called "swingy-ness." If a kobold with four hps can manage to survive for 10 rounds because of PC's bad rolls while doling out damage because of the DM's good rolls (and perhaps the kobold's tactics) that is a story not an annoyance!

The opposite is also true, the lucky hit that takes out a big bad monster in one or two hits is awesome. Most of the time stuff falls in the middle (or feels like it) and that is good enough for me, I don't need to do math and worry about if it is "consistent enough."

This is also true for PCs. The last standing and seriously weakened PC who manages to survive through a mix of luck and tactics is fun as hell!

As for monster hit points, I usually determine "average" (which for me is max hps for the first HD and then average from there - if the PCs get it the monsters get it! - though for 1 HD creatures I will tweak this down to a create a range of hps). Once I have that number, I think about when and where the PCs might face the creatures and decide if I want to tweak them up or down. If it is multiple monsters of a type I might make different sub-group with different HP totals. So yeah, on purpose in my games, one goblin might die if it takes 4 hps but another has 8. I like to downplay meta-game as much as possible, so I never want players to be thinking "this ogre took 30 hps of damage before dropping, so we need to do that much damage to his buddy." Maybe. But maybe he is weaker or stronger!

Narrating Hit Points: I do much like others have described, describing scrapes, hard numbing blows to a limb, desperate breath-taking dodges, headache inducing clangs against armor or a helmet, etc. . .I also describe how hurt creatures are based on a system of 4ths. . . down by up to one-quarter hps = lightly wounded, up to half = moderately wounded, up to three-quarters = seriously wounded, and from there to one HP remaining = critically wounded. I got these categories from previous edition healing spell names.

 

[Asisreo]

Some people mentioned Finger of Death versus Disintegrate (7th-level) and there's an incredibly important distinction that goes beyond Saving Throws and Expected Value.

That distinction is that while Disintegrate has a higher average, its standard deviation and variance is unusually high. This is because a Disintegrate that is successfully saved does 0 damage while Finger of Death still deals half-damage. The more reliable of the two is Finger of Death but an all-or-nothing attack is best done with Disintegrate.

This actually means that Finger of Death is useful in more scenarios than Disintegrate at its highest level, not mentioning that it can create unlimited zombies during downtime.

I calculated this with an Adult Black Dragon with a +10 to Con and a +7 to Dex so even when a creature has a high save and they're likelier to save with Con, its still better to use Finger of Death unless its within kill-range.

 

[fearsomepirate]

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.