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

[Asisreo]

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.

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.

I'm not necessarily trying to say that every action must be calculated before being able to use it. But with our mathematical model, we can find trends that allow us to make general judgements that we do not have to even think about.

We can understand that smaller but more numerous dice has a lower variance than larger but fewer dice. This means when comparing two attacks of similar damage, you'll want small dice for low-HP monsters and large dice for high-HP monsters.

So choosing whether to cast a 5th-level fireball versus a Cone of Cold comes down to whether the minions are generally strong or weak. If the minions are weak enough to die from a single fighter's hit, it's safe to say that Fireball is the correct choice. But if they take 3 or more hits, we can understand Cone of Cold is more reliable to kill, even if we don't actually have exact numbers.

 

[tetrasodium]

I think there is too much emphasis on smaller but more numerous dice vrs larger but fewer dice. The impact of how many attacks those dice are spread across and result of a failed attack play a huge role that shouldn't be ignored. Compare a level 5 or 11 fighter pally ranger or similar weapon wielder to a nonwarlock caster with an all or nothing cantrip or a single save for half leveled spell they wont simply recover next round. Those kinds of comparisons matter more than the extreme white room hypotheticals and play out side by side round after round session after session without the complications of things like " "but what if a rogue or monk uses a bonus action to x instead of y " that introduce a whole new set of non damage variables to weighing the best option

 

[Blue]

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.

This doesn't address the point I was bringing up about with more dice and needing a large number of hits to kill like happens constantly at higher levels in 5e. Can you present the math again but with a over a hundred HPs - Some CR 5 have that already. And where the HPs are not a perfect multiple of mean damage done.

 

[fearsomepirate]

All of this talk is why I pretty much ignore averages and look at distributions. Means are fine, but tails matter. How fat are the tails? Events with a 25% chance of happening matter. What's the chance a given thing does at least 35 damage?

Incidentally, this is the problem with the Wild Magic sorc. There's a 2% chance of Fireball on Self. This sounds tiny, but it means that a lot of people will experience it at least once at the levels where it means a TPK.

 

[fearsomepirate]

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.

Generally speaking, yes, it's more important to think about chance of ending a threat than average damage, but in this specific case, a creature at 19 hp is about 1% more likely to die from a 3d8 hit than a 3d6 hit. It's not a meaningful enough difference to plan actions around. That said, even small percentages matter...the difference between 0.1% and 0.01% is a factor of ten, after all.

 

[Sorcerers Apprentice]

Higher variance also means the chance of a creature surviving 2 hits is higher, you can't just look at the chance of a one hit kill in isolation.

 

[Asisreo]

Higher variance also means the chance of a creature surviving 2 hits is higher, you can't just look at the chance of a one hit kill in isolation.

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.

 

[Sorcerers Apprentice]

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.

Only if he knows that he needs around max damage to kill the monster. If he needs around average damage then the Greatsword has better odds of killing the target.

 

[fearsomepirate]

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.

Note that if you failed on round 1, you might want to switch up tactics on round 2, what with independent probabilities and all.

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.

In case of a barbarian, once Brutal Critical comes online, we also know that granting him advantage (or, even better, paralyzing an enemy for him) is that much more helpful if he is wielding a Greataxe.

 

[Asisreo]

Note that if you failed on round 1, you might want to switch up tactics on round 2, what with independent probabilities and all.

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.

Only if he knows that he needs around max damage to kill the monster. If he needs around average damage then the Great

In the absence of this information, though, its generally best to assume your target has more HP than the average of your attacks. That is, in this case, that the target of your attacks has 10 or more HP (assuming +3 STR).

Its probably safe to assume this to be the case unless you're fighting low fodder minions.

And yes, even with extra attack, though it would be 23 HP rather than 10 or 20HP as you may imagine.