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

[Asisreo]

You seem to be more fluent with the names of the math than I am, but I think you may be missing a point. I don't care if it EXACTLY hits the average because it's most likely not the case that they HP is an exact multiple of the attack damage. So if mean damage is 12 and the foe has a mean of 113 HPs, I have a high chance that ten hits will kill it. Because really I'm caring about tightness of the clustering.

And damaged absolutely is cumulative. Individual damage has absolutely no meaning, it's only in the cumulative where it's met or exceeded the total HPs of the foe does it matter. Looking at them as individual results is misleading because they have no meaning until they hit that threshold.

Damage is cumulative up to the target's HP, but stops being so afterwards.

Let's say a monster is a trial. Each trial is independent of each other and what you're basically asking is: how many dice rolls are needed to equal or exceed this monster's HP. Lets say you have a d6 damage and the monster has 6 HP.

Well, the RF of the trials never converges. That is to say: it doesn't matter how many times you've killed the monster before, this new monster is completely independent and it could take 6 rounds or it could take 1, the probability is random though 2 is the most likely amount of rounds.

Now, if you were to take a large amount of monsters and killed them all, you'd see the amount of rounds it takes to kill these monsters converge to 2. This is the CMF.

But you're not looking for the amount of rounds it takes for you to kill a larger amount of monsters, you're looking for the attack option that is likely to kill the one you're currently fighting faster.

You're looking for the attack with the highest probability to kill.

Its very rare for you to deal damage exactly equal to your enemy's HP. So when you look at your options, you should consider your probability to kill.

Let's say you're a wizard with firebolt but you also have a light crossbow with a +1 for dex meaning they have the same average damage (5.5). Lets say the enemy's AC is 0 (guaranteed to hit) and his HP is 7. A bit higher than either attack's expected damage. Which one should you choose or does it matter?

The answer is firebolt, which has a 40% chance to kill compared to the light crossbow's 37.5% chance to kill.


Now, here's an extreme example:

Let's compare 55(10d10) vs 56(2d6+49).

You might want to jump ahead and say that 2d6+49 is better than 10d10 by all accounts, but that's untrue.

If, for instance, the enemy has 57 HP, then its actually better to do 10d10 which has a 43.54% chance of killing rather than 2d6+49 which has a 41.67% chance of killing.

In this case, the option that does less expected damage has a higher probability of killing.

 

[jhingelshod]

Damage is cumulative up to the target's HP, but stops being so afterwards.

Let's say a monster is a trial. Each trial is independent of each other and what you're basically asking is: how many dice rolls are needed to equal or exceed this monster's HP. Lets say you have a d6 damage and the monster has 6 HP.

Well, the RF of the trials never converges. That is to say: it doesn't matter how many times you've killed the monster before, this new monster is completely independent and it could take 6 rounds or it could take 1, the probability is random though 2 is the most likely amount of rounds.

Now, if you were to take a large amount of monsters and killed them all, you'd see the amount of rounds it takes to kill these monsters converge to 2. This is the CMF.

But you're not looking for the amount of rounds it takes for you to kill a larger amount of monsters, you're looking for the attack option that is likely to kill the one you're currently fighting faster.

You're looking for the attack with the highest probability to kill.

Its very rare for you to deal damage exactly equal to your enemy's HP. So when you look at your options, you should consider your probability to kill.

Let's say you're a wizard with firebolt but you also have a light crossbow with a +1 for dex meaning they have the same average damage (5.5). Lets say the enemy's AC is 0 (guaranteed to hit) and his HP is 7. A bit higher than either attack's expected damage. Which one should you choose or does it matter?

The answer is firebolt, which has a 40% chance to kill compared to the light crossbow's 37.5% chance to kill.


Now, here's an extreme example:

Let's compare 55(10d10) vs 56(2d6+49).

You might want to jump ahead and say that 2d6+49 is better than 10d10 by all accounts, but that's untrue.

If, for instance, the enemy has 57 HP, then its actually better to do 10d10 which has a 43.54% chance of killing rather than 2d6+49 which has a 41.67% chance of killing.

In this case, the option that does less expected damage has a higher probability of killing.

You can make up examples like this that cut the other way.....what if the enemy had 51HP?

I get what you are saying....at least I think I do, and I apologise if I am putting words in your mouth....that Expectation (DPR) isn't the only important measure, but that's why statisticians have well developed models of central tendency. In reality, as has been pointed out, there are so many variables, measurable and unmeasurable, in a typical RP combat encounter that Std dev becomes a bit irrelevant, especially in a "real world" situation where the choice is between say 4d6 and 3d8 rather than 10d6 or 2d6+49.

 

[Asisreo]

You can make up examples like this that cut the other way.....what if the enemy had 51HP?

I get what you are saying....at least I think I do, and I apologise if I am putting words in your mouth....that Expectation (DPR) isn't the only important measure, but that's why statisticians have well developed models of central tendency. In reality, as has been pointed out, there are so many variables, measurable and unmeasurable, in a typical RP combat encounter that Std dev becomes a bit irrelevant, especially in a "real world" situation where the choice is between say 4d6 and 3d8 rather than 10d6 or 2d6+49.

Well, I'm trying to change the basis of exactly how we think of damage and HP rather than relying solely on DPR, expected outputs, and other unreliable measures. That there's far more complexity in the options you're given than pure DPR. I also enjoy math.

Lets use your 4d6 vs 3d8 example. The 4d6 option does an average of 14 damage while the 3d8 option does 13.5 damage. One clearly must have an advantage over the other, right?

Well, if you're fighting an enemy with HP less than 18 HP, you'd be correct. However, if you're fighting an enemy with more than 18 HP, its actually 3d8 damage that comes out on top. Even though they have the same maximum but 3d8 has a lower minimum and average, a higher HP creature is more likely to die from a single 3d8 than a single 4d6.

 

[Stalker0]

Well, if you're fighting an enemy with HP less than 18 HP, you'd be correct. However, if you're fighting an enemy with more than 18 HP, its actually 3d8 damage that comes out on top. Even though they have the same maximum but 3d8 has a lower minimum and average, a higher HP creature is more likely to die from a single 3d8 than a single 4d6.

I'm going to clarify your example. You are saying that in the scenario where the monster has 18-24 hp (I am presuming), that a 3d8 attack is more likely to kill than a 4d6 one. Which is true.

But again, these corner case examples ignore the simple fact that rarely are players able to adjust their damage values so aggressively, and rarely have enough in game knowledge (aka the monster's exact HP), that they can make such decisions for maximum gain.

So instead of trying to optimize your damage routine for every round and every scenario (which is nigh impossible to do)....the general optimization is a higher average DPR. Sure on occasion the lower average with the higher variance will net you a kill you wouldn't have gotten otherwise. Or...you roll a really high result on a high variance die....when you only needed a single point of damage.

So I'm not really sure what this mathematical knowledge buys us?

 

[Benjamin Olson]

I think the bigger elephant in the room for damage calculations with Goblin killing (or other low hp enemy) is not whether one hit actually kills them as much as the average damage might indicate, but rather the fact that so much of your damage output is likely to be useless overkill damage. Especially when they actually do manage to survive the first strike they are likely hanging on at 1-3 hp, and your follow up is going to usually (almost certainly if you have a set modifier to the die) waste some potential damage.

As for this rolling HP issue, I will say that goblins and the like are the only ones for whom I actually routinely do roll the HP. Goblins, specifically, are my favorite for this as each is just two d6s without a modifier, and that is the level of math where I can look and just know the total rather than having to apply any thought to it. With one, two, sometimes three hit die creatures it can actually make a substantial difference and make combat more interesting. It's when you get beyond that it starts becoming statistically less and less likely to get any major difference, while at the same time the rolling process becomes more and more cumbersome.

 

[Ovinomancer]

I think the bigger elephant in the room for damage calculations with Goblin killing (or other low hp enemy) is not whether one hit actually kills them as much as the average damage might indicate, but rather the fact that so much of your damage output is likely to be useless overkill damage. Especially when they actually do manage to survive the first strike they are likely hanging on at 1-3 hp, and your follow up is going to usually (almost certainly if you have a set modifier to the die) waste some potential damage.

As for this rolling HP issue, I will say that goblins and the like are the only ones for whom I actually routinely do roll the HP. Goblins, specifically, are my favorite for this as each is just two d6s without a modifier, and that is the level of math where I can look and just know the total rather than having to apply any thought to it. With one, two, sometimes three hit die creatures it can actually make a substantial difference and make combat more interesting. It's when you get beyond that it starts becoming statistically less and less likely to get any major difference, while at the same time the rolling process becomes more and more cumbersome.

I did a long bit on damage overkill in DPR calculations in one of these threads last year. It has a noticeable impact, but it's not that big. Usually a few points off the top when damage is high.

However, that acknowledged, time to kill is the better metric -- overkill really only matters if you're looking at DPS calcuations, which really don't say much about TTK. High damage does tend to correlate well with TTK, so overkill can be a red herring depending on the discussion.

 

[Asisreo]

I'm going to clarify your example. You are saying that in the scenario where the monster has 18-24 hp (I am presuming), that a 3d8 attack is more likely to kill than a 4d6 one. Which is true.

But again, these corner case examples ignore the simple fact that rarely are players able to adjust their damage values so aggressively, and rarely have enough in game knowledge (aka the monster's exact HP), that they can make such decisions for maximum gain.

So instead of trying to optimize your damage routine for every round and every scenario (which is nigh impossible to do)....the general optimization is a higher average DPR. Sure on occasion the lower average with the higher variance will net you a kill you wouldn't have gotten otherwise. Or...you roll a really high result on a high variance die....when you only needed a single point of damage.

So I'm not really sure what this mathematical knowledge buys us?

It buys us the ability to make more informed decisions about our game plan as players and DMs. Knowing how manipulating the dice, bonuses, and accuracy of our attacks can allow us to make better, more optimized decisions.

Even moreso as a DM since you can use this knowledge to discreetly buff or nerf your encounters, making them harder or easier depending on how you manipulate the averages and variances.

But also, players do have the ability to closely control how they can go about their damage, they are usually too concerned about DPR to get a good sense of how they can change their tactics.

For example, I'll use a previous character of mine: a Ranger with +2 STR and +3 DEX. She wielded a battleaxe or a shortsword with a shield. Now, typically DPR would say that it doesn't matter which weapon I use, but the dice actually tells us that any creature below 7 HP should be attacked with the shortsword and all other creatures should be attacked with the battleaxe (one-handed).

This can be very important, especially at lower levels where any attack can prevent a teammate kill.

Players can also get a good estimation on what their health is by the fact that the DM (is supposed to) describe a creature below half-health as having visible bruises and cuts. Its not completely accurate but it gives a nice picture.

 

[Ovinomancer]

It buys us the ability to make more informed decisions about our game plan as players and DMs. Knowing how manipulating the dice, bonuses, and accuracy of our attacks can allow us to make better, more optimized decisions.

Even moreso as a DM since you can use this knowledge to discreetly buff or nerf your encounters, making them harder or easier depending on how you manipulate the averages and variances.

But also, players do have the ability to closely control how they can go about their damage, they are usually too concerned about DPR to get a good sense of how they can change their tactics.

For example, I'll use a previous character of mine: a Ranger with +2 STR and +3 DEX. She wielded a battleaxe or a shortsword with a shield. Now, typically DPR would say that it doesn't matter which weapon I use, but the dice actually tells us that any creature below 7 HP should be attacked with the shortsword and all other creatures should be attacked with the battleaxe (one-handed).

This can be very important, especially at lower levels where any attack can prevent a teammate kill.

Players can also get a good estimation on what their health is by the fact that the DM (is supposed to) describe a creature below half-health as having visible bruises and cuts. Its not completely accurate but it gives a nice picture.

Ooh, not so sure about your axe/short sword delta there. I mean, going for the higher possible outcome while keeping the mean seems good, but, in reality, the 5% delta in hit chances eats pretty strongly into the difference in the high end and lowers the low end further, and makes the mean damage a touch less. This is maximized at when you need a 20 to hit with the axe (19 with the sword) and actually evens out when you need a 1 or less for both (so a 2+ hits). These are the extremes, but in the middle (.5 for sword, .45 for axe), the top end of the d8+2 is reduced to exactly the same as the d6+3, while the bottom of the d8 is reduced below the d6 even further.

 

[pemerton]

Its very rare for you to deal damage exactly equal to your enemy's HP. So when you look at your options, you should consider your probability to kill.

Let's say you're a wizard with firebolt but you also have a light crossbow with a +1 for dex meaning they have the same average damage (5.5). Lets say the enemy's AC is 0 (guaranteed to hit) and his HP is 7. A bit higher than either attack's expected damage. Which one should you choose or does it matter?

The answer is firebolt, which has a 40% chance to kill compared to the light crossbow's 37.5% chance to kill.

Sure. But how does a player know the goblin's hit points? If the target is instead a goblin or kobold with 4 hp, then the crossbow is better because with 1d8+1 only 1/4 of rolls are below 4, whereas with 1d10 3/10 of rolls are below 4.

In any event I don't think @Blue was thinking of fights against goblins. The point was that in fights at mid- and upper levels, the players (via their PCs) defeat monsters by piling on the results of multiple damage dice rolls (from multiple attacks, spells etc) at which point there is a general trend away from extreme results and towards mean-ish ones. At which point the practical difference between (say) 1d10 and 1d8+1 tends to reduce.

players do have the ability to closely control how they can go about their damage, they are usually too concerned about DPR to get a good sense of how they can change their tactics.

Leaving aside the playtime overhead of doing this sort of calculation, there is a mechanical overhead also - eg your wizard switching from spellcasting (with a focus?) to a crossbow has to engage the "changing held/wielded object" rules, which in turn impact the action economy.

My feeling is that for this sort of thing to be worthwhile the optimisation benefits have to be bigger and more obvious. Eg in my long-running 4e game the fighter would switch between a really big axe (good damage) and a not-quite-as-buff polearm (lesser damage, but superb reach). The player also took proactive steps in making build choices in order to manage the action economy implications of changing his weapon from round to round or even sometimes within a round.

 

[Sword of Spirit]

What I learned from this is that a 1d12 weapon might be just as effective as a 2d6 weapon in practice. (If you don‘t include the Great Weapon Fighting Style which is a lot better for 2d6.)