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

#### Asisreo

So, I see very often DPR, Damage Comparisons, and Health being discussed at-length playing D&D.

Often times, in White-Room theory-crafting scenarios, someone will talk about the damage a character can inflict on a target and compare that damage to the health of said target.

For example, someone could talk about how a single scorching ray kills a goblin because 2d6 = 7 average damage and a goblin's average HP is 7. Therefore, if the ray hits, its essentially a guaranteed kill, right?

But we're forgetting the fact that when average damage = 7, it actually means there's only a 58.33% chance to actually kill that enemy. This is because while 7 is the most likely sum of combinations, it still only accounts for 16.66% of the total possible combinations.

So if you have a 65% chance to-hit a goblin with 2d6 damage, you actually only have a 38% chance of killing the goblin, which is really low if you're taking a whole action. Its possible, but its very low.

Now, reverse that but for HP. Imagine the DM decided he wanted to roll for health but he waited until after the grimlock first takes damage. The damage is rolled and it ends up 11, the DM decides to roll the grimlock's health which is 2d8+2. The damage should kill, right? Well, its actually around 50% as well.

If you combine those two mathematical models, the actual percent chance of certain attacks killing a character with rolled dice becomes much swingier. Of course, most DM's don't roll health for their monsters, but it does lead to interesting probabilities.

I just wanted to discuss exactly how damage can be a misleading factor when talking about damage and its relation to HP.

TL;DR
When average damage = average health, it isn't a guaranteed kill. Its actually roughly a 50% chance to kill. Be considerate of these facts when discussing DPR.

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#### aco175

##### Legend
Do you think some of the theory discussion involves average damage and average HP? Some of the other threads we talk about has people saying that monsters should only deal average damage to speed things up or aid the DM. Not sure how many players use average damage over rolling the damage.

I'm not big on math so I'm not sure how much of this thread will evolve to be over my head, but it sounds worth talking about.

#### Oofta

##### Legend
Not a math geek (at all) but I use average HD and average damage for monsters (except crits) because it's easier. I even allow players to use average damage* for those that can't do quick math in their heads.

But I personally don't pay a lot of attention to white room analysis because it tends to focus too much on offense (and combat for that matter), do not pay enough attention to flexibility or defenses and the differences most people see are so minor you'll never notice it in an actual game.

Anyway carry on, I'll be interested to see what comes of the discussion, even if I only understand a third of it. That's understanding .25%, right?

*Rounded up for PC damage, because I don't want them to feel penalized.

#### fearsomepirate

##### Hero
Not only is that mean damage, but chance to hit on a goblin isn't all that high at 2nd level. A typical caster will have +4 or +5 to attack, and a goblin has 15 AC. If you take the higher of the two values, your chance to kill the goblin with single ray is 34%. With the lower, it's 31%.

#### jasper

##### Rotten DM
I am an Adventure League dm. I use avg damage and hp most of time. Occasionally I use Avg damage on out going spells.
Back in 1e I generally did roll health for my monsters. Some times it got wacky.

#### DND_Reborn

##### The High Aldwin
FWIW considering average damage and hp:

As DM, I've always used average HP for monsters/NPCs except BBEG who get max HP. Damage is also average, except if average damage would automatically down a PC, in which case I will roll to give the PC a chance to remain in the fight.

For the players, about 80% of the time, they just accept the "average" HP when they level instead of rolling for hit points. Players can also just use average damage for their attacks and spells (like just using 28 for a fireball instead of rolling). Players use average damage probably about 80% also. shrug

#### Quartz

##### Hero
Imagine the DM decided he wanted to roll for health but he waited until after the grimlock first takes damage.

I don't know about other GMs but where possible I pre-roll monster HPs or just invoke the mook rule - one hit (or two / three / whatever hits) and they're dead.

#### Snarf Zagyg

##### Notorious Liquefactionist
So, I see very often DPR, Damage Comparisons, and Health being discussed at-length playing D&D.
(snip)

I just wanted to discuss exactly how damage can be a misleading factor when talking about damage and its relation to HP.

IME, basic math can be a useful tool when it comes to someone looking to compare things in isolation (aka, optimization or efficiency).

To use an easy example, if everything else is equal, what does more damage- 2d6 or 1d12? Easy, right? So if you're presented with an option for damage, with everything else being equal, you would choose the 2d6 option.

It's the same in many fields; we've seen increased reliance on these metrics in sports. The value of a three point shot will be higher than a two point shot (of course), and then you can look at the expected field goal percentage (the chance of "hitting" to use a D&D term) to see the expected "damage" from each shot (the expected points). Which is why, in basketball, a lot of teams now play for either the three-point shot or the dunk/close two, and eschew the long-range two point shot.*

Which gets to two separate issues:

1. What does white room theory have to say about any individual combat? Or, as you put it, how does it account for the "swinginess" of dice? And the answer is- it doesn't. Not at all. The process matters more than the results. Think of it like this by analogy; if someone shoots 40 from the three point line, and 50 percent from within the arc (two points), then they should take the three point shot (1.2 expected from 3, 1 from 2). Even if they happen to miss their three pointer because of swinginess, it was still the correct decision- the theory was correct.

2. On the other hand, as many people point out, white room theory in D&D is often flawed. There is a lot of bad math. There is a failure to account for tradeoffs (AC, health, other effects) in combat. And it does not even try to math out "out of combat" effect in D&D. In short, from what I have seen, it tends to be very limited outside of comparing like things, such as DPR, without full context. No one to my knowledge has done a good, holistic, comprehensive statistic like "WAR" (from baseball) for D&D.

*Of course, as defenses shift, there is now a new efficiency in long twos, but that's a different issue.

#### turnip_farmer

So if you have a 65% chance to-hit a goblin with 2d6 damage, you actually only have a 38% chance of killing the goblin, which is really low if you're taking a whole action. Its possible, but its very low.
You get three scorching rays per one action. So your theoretical wizard will kill at least one goblin a large majority of the time; and has a little over a 5% chance of taking out three in one go.

#### Asisreo

You get three scorching rays per one action. So your theoretical wizard will kill at least one goblin a large majority of the time; and has a little over a 5% chance of taking out three in one go.
To be precise, if you separated the rays at three individual goblins, there's a 76% chance to kill at least one.

In contrast, there's a 5.48% chance of killing all three.

If you were to point them at an individual goblin, it would be a bit tougher to precisely calculate...

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