More of a math question really, hope this is the right place.
Trying to settle a dispute about whether there is average damage difference (or any significant imbalance at all) between a single attack that deals all the damage vs multiple attacks with the same total damage output.
Ex.
1 attack roll the deals 5d8 vs 5 attack rolls the deal 1d8.
not too concerned with crits and there is no ability modifier or anything else to the damage. Just dice.
Depends on Resistance and Defenses. For High Defense Targets, Multiple Attacks are better because you would get more chances to hit in the first place. If the target has Resistance then more initial damage is better since you always round down. Now there's also a issue of which specific dice are being used, this is the difference between a 1d12 and a 2d6. Both can be 12, but 1d12 has a 1/12th chance while 2d6 has a 1/36th chance. If that sounds like a 1d12 is better NOPE! A 1d12 also has a 1/12th chance of getting a 1 or a 2, while a 2d6 can not get a 1 and has only a 1/36th chance of getting a 2. If the maximum possible is the same: Rolling more Dice gives more average rolls, Rolling less Dice gives both higher and lower potential.
TL;DR: It is completely situational, there is no optimal choice between the two.
I don’t want to know which is better - I want to know IF there even is a ‘better’. My suspicion is that there is no ‘better’ and that if you had to make the choice between the two as a PC it would mostly be a matter of preference and not ‘clear mechanical advantage’.
Yeah basically my DM suggest that multiple attacks is default worse than a single attack and I disagree thinking there is no inherent advantage to either (outside of circumstantial advantage).
My DM thinks that the multiple attack option would need to have its damage increased over the single attack alternative in order to stay “competitive” and I think that’s totally false.
The expected damage is the same for 1 attack for 5d8 vs 5 attacks for 1d8.
If you are attacking creatures with low hit points, 5d8 is probably overkill. With 5 attacks, you can potentially take down 5 kobolds. One attack for 5d8 will just kill one but it will be very dead. Maybe your DM will help you out by describing the kobold as being cut in half by the fearsome blow and the other kobolds flee in terror. Instant death is going to be more likely with 5d8, which makes it worse for PCs than five 1d8 attacks. Most DMs aren’t going to keep attacking you after you drop to 0 hp.
the issue is more about rolling once to hit vs rolling several times to hit. the specific damage dice matter little, as the raw averages on x dN vs dN x times are the same. Generally, you prefer more chances to hit. what might affect calculations is the number of times a damage bonus could be added to the multiple attacks, which could add a significant static modifier to the average damage results
Considering attacks that add up to the same type and amount of dice (e.g. 5 attacks that do 1d8 each vs 1 attack that does 5d8):
Mathematically, against a giant bag of hit points, there is very little difference. There is a very slight advantage to the one big attack, if there is damage resistance involved: the big attack will never do 0 damage, while the small attacks could all do 0 damage, if they roll 1's (which get halved and rounded down to 0). That goes away if there is any modifier, though.
There is a difference, though, when the target has a relatively small amount of hit points (relative to the damage). Like Trirhabda mentioned, with the small attacks you could potentially down several targets. On the other hand, if the game is using the massive damage rules, you're more likely to automatically kill (without reducing to 0 hps) a target with the one big attack.
(In other systems, where damage resistance/reduction works by subtracting a specific amount from damage, rather than a percentage, big attacks have an edge, since they'll be reduced proportionally less than small attacks, but that's not the case in D&D 5e.)
When attacks have similar average, min, and/or max damage, but different dice types/amounts, it's different. For example, 1d12 vs 2d6, the single d12 will have a slightly lower average damage (6.5, vs the 7 of the 2d6), and a higher variance, than the 2d6. 2d6 will usually be considered "better", because of it's slightly higher average, but mostly because of it's more dependable damage (it will roll closer to average more frequently), so it's easier to plan around. You will get higher numbers more frequently with 1d12, though, so some players prefer that. Of course, there's a significant difference when abilities that allow you to roll an extra die on crits is involved.
It is complicated as the previous answers have shown.
The thing is a 5d8 attack can either be all or nothing based on whether the hit is made (ignoring limited resistances), while 1d8 by 5 attacks can be anything from 1d8 damage up to 5d8. Alot of this will really come down to how easy the target is to hit, and whether the 5 attacks are in one round or over multiple rounds.
If the 5d8 or 5 attacks are in a single round there isn't much difference in effectiveness, over multiple rounds the instant 5d8 could eliminate the threat for consecutive rounds where a few hits would not.
While if a target is easy to hit the odds are in favour of hitting with one attack but potentially missing with one of the 5 attacks, thus the 5 doing less damage overall. However if the target is hard to hit, the potential is for the one attack to miss completely (doing no damage) while the odds are in favour to hit at-least one of the 5 attacks, thus doing some damage.
At the end of the day overall its much the same, but different situations make either option better or worse.
As above, they are equal in average damage. Most differences are only situational.
For 5 attacks d8: pros; divisible to kill more small enemies, less chance to do zero damage, any buff to all attacks will multiply. cons; almost no chance to deliver max damage for a turn, debuffs that affect all attacks could multiply.
For 1 attack 5d8: pros; ok chance exists of massive damage crit to take down big guy early, buff or nova-ability targeting single attack per turn at most effective. cons; awkwardly too powerful for rat killing, 'disadvantage on next attack' or similar debuff has more effect, significant chance of zero damage in a turn.
The big thing I can think of to consider is the attack of opportunity. All creatures can use that reaction, but they only ever make one melee weapon attack (even if they have multi-attack of any sort on their own turn). One of these creatures has a 1d8 attack of opportunity, while the other has 5d8. That's a very significant difference which may affect battlefield behaviour.
All up though, I think those pros and cons all even out to no significant difference either way.
My DM won’t read this thread as he think you’re all ‘armchair mathematicians’ and he ran his own calculations to convince himself that a single attack for 5d8 has a statistical advantage over multiple attacks in one round each dealing 1d8.
The full story (short version) is that we are designing 2 home brew spells and a player can only pick one if them: single melee spell att that deals 5d8 vs 5 ranged spell att that deal 1d8 each.
We want the damage to be balanced but he thinks that means we have to increase the number of ranged attacks to increase its maximum over the single melee.
And he’s convinced math is on his side and won’t budge unless someone can say explicitly, mathematically, by what principle he is wrong.
He convinced himself by running some coin flip probability web app like any true math genius.
Any takers want to prove my DM wrong? (If he is wrong, that is, I have no idea I’m an idiot I just disagree with him and think he’s getting to big for his britches)
As pretty much all of us have shared, there are cases where one is better than the other, but it is all edge cases that don't matter to overall balance. If your DM thinks that out of a Greatsword (2d6) or Greataxe (1d12) one is objectively better than the other, then he has a personal preference that he has allowed to become an absolute. You probably can not reason with him.
However, it should be noted that when attacks gain Ability Score Modifiers to them a 5d8+Cha Mod will probably do less than a 1d8+Cha Mod done 5 times. This is because in the first example the Charisma Modifier is added once, while in the second example the Charisma Modifier is added 5 times. At maximum possible damage (with 20 Charisma, which is a +5 modifier) this is 45 Damage versus 65 Damage. Meaning that multiple attacks are better if positive Modifiers are added.
If that doesn't convince him, nothing will. Like Albert Einstein he will deny Quantum Mechanics to his Death Bed. Like all the foolish Mathematicians before him, he is wasting time trying to prove Euclid's Fifth Postulate.
Edit: If it helps any, I'm an Accounting Major. While I may not do Calculus, Math is kind of my life.
My DM won’t read this thread as he think you’re all ‘armchair mathematicians’ and he ran his own calculations to convince himself that a single attack for 5d8 has a statistical advantage over multiple attacks in one round each dealing 1d8.
The full story (short version) is that we are designing 2 home brew spells and a player can only pick one if them: single melee spell att that deals 5d8 vs 5 ranged spell att that deal 1d8 each.
We want the damage to be balanced but he thinks that means we have to increase the number of ranged attacks to increase its maximum over the single melee.
And he’s convinced math is on his side and won’t budge unless someone can say explicitly, mathematically, by what principle he is wrong.
He convinced himself by running some coin flip probability web app like any true math genius.
Any takers want to prove my DM wrong? (If he is wrong, that is, I have no idea I’m an idiot I just disagree with him and think he’s getting to big for his britches)
Um... should be easy. Let's see.
In a single turn, assuming every attack hits, both attacks do the same amount of damage: 5d8. In one case, it's a single big attack, while in the other it's five smaller attacks, but the damage rolled is the same: 5d8. The minimum damage of each is the same (5), the maximum damage is the same (40), the average is the same (22.5), and, in fact, the entire distribution curve for both rolls is the same, because they're the same roll.
Let's call the "5d8" damage a, and the "1d8" b. The single big attack does a damage, and the multiple smaller ones do b damage, each, for a total of 5 * b damage. Obviously, a = 5 * b, so both attacks do the same damage, whether you want to call it a or 5 * b.
Now, let's not assume every attack will hit. Let's say attacks hit with a probability we'll call h. If you have a total bonus to hit of +5, and you're trying to hit something with AC 16; in that case, you'd need an 11 or higher on a d20 to hit, which means you hit 50% of the time, and h would equal 0.5. If you had a total bonus to hit of +8, and were trying to hit something with AC 12, you'd hit 85% of the time, and h would equal 0.85. If you use the big attack each turn for n turns, you can expect to hit h * n times (e.g. if you have a 50% chance to hit, and you attack 10 times, you can expect to hit 5 times). Each hit does a damage, so the total expected damage would be h * n * a. If you use to smaller attack every turn for n turns, you can expect to hit h * n * 5 times (note the "times 5"; the "small" attack is 5 attacks per turn, so over n turns, you'd be rolling n * 5 attacks). Each attack does b damage, so the total expected damage would be h * n * 5 * b. But we've established that a = 5 * b, so h * n * a = h * n * 5 * b, meaning both attacks do the same damage, for any value of h, over any number of turns n. Q.E.D.
... and some of us are likely not "armchair" mathematicians but actually do math for a living.
Average damage on a d8 is 4.5
Average damage on 5d8 rolled at once is 5 x 4.5 = 22.5
Average damage on d8 rolled 5 times separately IS (surprise (not)) = 4.5 + 4.5 + 4.5 +4.5 + 4.5 = 22.5
Both 5d8 rolled at once and 5 d8s rolled consecutively have EXACTLY the same average value.
Now lets factor in the to-hit probability (we can call it h like Tonio :) ).
The to hit probability is the chance of rolling greater than your target number with all your modifiers. If you need to roll a 15 or more to hit then your to hit probability is 30% ... 15 or 16 or 17 or 18 or 19 or 20 = 6 chances out of 20 = 3/10 = 30%.
So if I roll one die for my 5d8 the average damage is h * 5d8 + (1-h) * zero ... we can ignore the second part since it is ZERO = h * 5d8
If I roll 5 to hit die, one for each d8 then the average damage is h*d8 + h*d8 + h*d8 + h*d8 + h*d8 ... and GUESS what? This is exactly the same as h * 5d8.
SO ... no matter which way it is calculated the AVERAGE damage of each is EXACTLY the same whether you roll it all at once or separately ... and if your DM wants to stick by his incorrect understanding then that is fine but you can just know he is wrong :)
------------------
So ... WHAT is the difference between these if they have exactly the same average damage?
1) Rolling 5 separate dice will do the same average damage over time but will do less damage more often.
2) There is only one circumstance where they will do different damage and that is when resistance is involved.
- 5 d8 rolled separately do 4/2 damage each for an average of 10 damage
- 5d8 rolled at once does 22.5/2 damage for an average of 11
So the only time these do different amounts of damage is when resistance is involved and even then the difference is very small and favors rolling all the dice at once.
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More of a math question really, hope this is the right place.
Trying to settle a dispute about whether there is average damage difference (or any significant imbalance at all) between a single attack that deals all the damage vs multiple attacks with the same total damage output.
Ex.
1 attack roll the deals 5d8 vs 5 attack rolls the deal 1d8.
not too concerned with crits and there is no ability modifier or anything else to the damage. Just dice.
Lemne know if I need to provide more details.
thanks in advance.
Depends on Resistance and Defenses. For High Defense Targets, Multiple Attacks are better because you would get more chances to hit in the first place. If the target has Resistance then more initial damage is better since you always round down. Now there's also a issue of which specific dice are being used, this is the difference between a 1d12 and a 2d6. Both can be 12, but 1d12 has a 1/12th chance while 2d6 has a 1/36th chance. If that sounds like a 1d12 is better NOPE! A 1d12 also has a 1/12th chance of getting a 1 or a 2, while a 2d6 can not get a 1 and has only a 1/36th chance of getting a 2. If the maximum possible is the same: Rolling more Dice gives more average rolls, Rolling less Dice gives both higher and lower potential.
TL;DR: It is completely situational, there is no optimal choice between the two.
I don’t want to know which is better - I want to know IF there even is a ‘better’. My suspicion is that there is no ‘better’ and that if you had to make the choice between the two as a PC it would mostly be a matter of preference and not ‘clear mechanical advantage’.
Yeah basically my DM suggest that multiple attacks is default worse than a single attack and I disagree thinking there is no inherent advantage to either (outside of circumstantial advantage).
My DM thinks that the multiple attack option would need to have its damage increased over the single attack alternative in order to stay “competitive” and I think that’s totally false.
Am I off my rocker?
No, you are correct.
There is a slight difference is certain situations, but both are equivalent overall.
Your DM is probably doing a little of the mental exercise I just did, but forgot the other half.
There is no better, there is no optimal, it's just a preference based on which logical fallacy you like more.
I like to roll more, it feels nice.
Some like to roll big, it feels nice.
The expected damage is the same for 1 attack for 5d8 vs 5 attacks for 1d8.
If you are attacking creatures with low hit points, 5d8 is probably overkill. With 5 attacks, you can potentially take down 5 kobolds. One attack for 5d8 will just kill one but it will be very dead. Maybe your DM will help you out by describing the kobold as being cut in half by the fearsome blow and the other kobolds flee in terror. Instant death is going to be more likely with 5d8, which makes it worse for PCs than five 1d8 attacks. Most DMs aren’t going to keep attacking you after you drop to 0 hp.
the issue is more about rolling once to hit vs rolling several times to hit. the specific damage dice matter little, as the raw averages on x dN vs dN x times are the same. Generally, you prefer more chances to hit. what might affect calculations is the number of times a damage bonus could be added to the multiple attacks, which could add a significant static modifier to the average damage results
Considering attacks that add up to the same type and amount of dice (e.g. 5 attacks that do 1d8 each vs 1 attack that does 5d8):
Mathematically, against a giant bag of hit points, there is very little difference. There is a very slight advantage to the one big attack, if there is damage resistance involved: the big attack will never do 0 damage, while the small attacks could all do 0 damage, if they roll 1's (which get halved and rounded down to 0). That goes away if there is any modifier, though.
There is a difference, though, when the target has a relatively small amount of hit points (relative to the damage). Like Trirhabda mentioned, with the small attacks you could potentially down several targets. On the other hand, if the game is using the massive damage rules, you're more likely to automatically kill (without reducing to 0 hps) a target with the one big attack.
(In other systems, where damage resistance/reduction works by subtracting a specific amount from damage, rather than a percentage, big attacks have an edge, since they'll be reduced proportionally less than small attacks, but that's not the case in D&D 5e.)
When attacks have similar average, min, and/or max damage, but different dice types/amounts, it's different. For example, 1d12 vs 2d6, the single d12 will have a slightly lower average damage (6.5, vs the 7 of the 2d6), and a higher variance, than the 2d6. 2d6 will usually be considered "better", because of it's slightly higher average, but mostly because of it's more dependable damage (it will roll closer to average more frequently), so it's easier to plan around. You will get higher numbers more frequently with 1d12, though, so some players prefer that. Of course, there's a significant difference when abilities that allow you to roll an extra die on crits is involved.
It is complicated as the previous answers have shown.
The thing is a 5d8 attack can either be all or nothing based on whether the hit is made (ignoring limited resistances), while 1d8 by 5 attacks can be anything from 1d8 damage up to 5d8. Alot of this will really come down to how easy the target is to hit, and whether the 5 attacks are in one round or over multiple rounds.
If the 5d8 or 5 attacks are in a single round there isn't much difference in effectiveness, over multiple rounds the instant 5d8 could eliminate the threat for consecutive rounds where a few hits would not.
While if a target is easy to hit the odds are in favour of hitting with one attack but potentially missing with one of the 5 attacks, thus the 5 doing less damage overall.
However if the target is hard to hit, the potential is for the one attack to miss completely (doing no damage) while the odds are in favour to hit at-least one of the 5 attacks, thus doing some damage.
At the end of the day overall its much the same, but different situations make either option better or worse.
- Loswaith
As above, they are equal in average damage. Most differences are only situational.
For 5 attacks d8: pros; divisible to kill more small enemies, less chance to do zero damage, any buff to all attacks will multiply. cons; almost no chance to deliver max damage for a turn, debuffs that affect all attacks could multiply.
For 1 attack 5d8: pros; ok chance exists of massive damage crit to take down big guy early, buff or nova-ability targeting single attack per turn at most effective. cons; awkwardly too powerful for rat killing, 'disadvantage on next attack' or similar debuff has more effect, significant chance of zero damage in a turn.
The big thing I can think of to consider is the attack of opportunity. All creatures can use that reaction, but they only ever make one melee weapon attack (even if they have multi-attack of any sort on their own turn). One of these creatures has a 1d8 attack of opportunity, while the other has 5d8. That's a very significant difference which may affect battlefield behaviour.
All up though, I think those pros and cons all even out to no significant difference either way.
My DM won’t read this thread as he think you’re all ‘armchair mathematicians’ and he ran his own calculations to convince himself that a single attack for 5d8 has a statistical advantage over multiple attacks in one round each dealing 1d8.
The full story (short version) is that we are designing 2 home brew spells and a player can only pick one if them: single melee spell att that deals 5d8 vs 5 ranged spell att that deal 1d8 each.
We want the damage to be balanced but he thinks that means we have to increase the number of ranged attacks to increase its maximum over the single melee.
And he’s convinced math is on his side and won’t budge unless someone can say explicitly, mathematically, by what principle he is wrong.
He convinced himself by running some coin flip probability web app like any true math genius.
Any takers want to prove my DM wrong? (If he is wrong, that is, I have no idea I’m an idiot I just disagree with him and think he’s getting to big for his britches)
As pretty much all of us have shared, there are cases where one is better than the other, but it is all edge cases that don't matter to overall balance. If your DM thinks that out of a Greatsword (2d6) or Greataxe (1d12) one is objectively better than the other, then he has a personal preference that he has allowed to become an absolute. You probably can not reason with him.
However, it should be noted that when attacks gain Ability Score Modifiers to them a 5d8+Cha Mod will probably do less than a 1d8+Cha Mod done 5 times. This is because in the first example the Charisma Modifier is added once, while in the second example the Charisma Modifier is added 5 times. At maximum possible damage (with 20 Charisma, which is a +5 modifier) this is 45 Damage versus 65 Damage. Meaning that multiple attacks are better if positive Modifiers are added.
If that doesn't convince him, nothing will. Like Albert Einstein he will deny Quantum Mechanics to his Death Bed. Like all the foolish Mathematicians before him, he is wasting time trying to prove Euclid's Fifth Postulate.
Edit: If it helps any, I'm an Accounting Major. While I may not do Calculus, Math is kind of my life.
Um... should be easy. Let's see.
In a single turn, assuming every attack hits, both attacks do the same amount of damage: 5d8. In one case, it's a single big attack, while in the other it's five smaller attacks, but the damage rolled is the same: 5d8. The minimum damage of each is the same (5), the maximum damage is the same (40), the average is the same (22.5), and, in fact, the entire distribution curve for both rolls is the same, because they're the same roll.
Let's call the "5d8" damage a, and the "1d8" b. The single big attack does a damage, and the multiple smaller ones do b damage, each, for a total of 5 * b damage. Obviously, a = 5 * b, so both attacks do the same damage, whether you want to call it a or 5 * b.
Now, let's not assume every attack will hit. Let's say attacks hit with a probability we'll call h. If you have a total bonus to hit of +5, and you're trying to hit something with AC 16; in that case, you'd need an 11 or higher on a d20 to hit, which means you hit 50% of the time, and h would equal 0.5. If you had a total bonus to hit of +8, and were trying to hit something with AC 12, you'd hit 85% of the time, and h would equal 0.85. If you use the big attack each turn for n turns, you can expect to hit h * n times (e.g. if you have a 50% chance to hit, and you attack 10 times, you can expect to hit 5 times). Each hit does a damage, so the total expected damage would be h * n * a. If you use to smaller attack every turn for n turns, you can expect to hit h * n * 5 times (note the "times 5"; the "small" attack is 5 attacks per turn, so over n turns, you'd be rolling n * 5 attacks). Each attack does b damage, so the total expected damage would be h * n * 5 * b. But we've established that a = 5 * b, so h * n * a = h * n * 5 * b, meaning both attacks do the same damage, for any value of h, over any number of turns n. Q.E.D.
I agree with Tonio :)
... and some of us are likely not "armchair" mathematicians but actually do math for a living.
Average damage on a d8 is 4.5
Average damage on 5d8 rolled at once is 5 x 4.5 = 22.5
Average damage on d8 rolled 5 times separately IS (surprise (not)) = 4.5 + 4.5 + 4.5 +4.5 + 4.5 = 22.5
Both 5d8 rolled at once and 5 d8s rolled consecutively have EXACTLY the same average value.
Now lets factor in the to-hit probability (we can call it h like Tonio :) ).
The to hit probability is the chance of rolling greater than your target number with all your modifiers. If you need to roll a 15 or more to hit then your to hit probability is 30% ... 15 or 16 or 17 or 18 or 19 or 20 = 6 chances out of 20 = 3/10 = 30%.
So if I roll one die for my 5d8 the average damage is h * 5d8 + (1-h) * zero ... we can ignore the second part since it is ZERO = h * 5d8
If I roll 5 to hit die, one for each d8 then the average damage is h*d8 + h*d8 + h*d8 + h*d8 + h*d8 ... and GUESS what? This is exactly the same as h * 5d8.
SO ... no matter which way it is calculated the AVERAGE damage of each is EXACTLY the same whether you roll it all at once or separately ... and if your DM wants to stick by his incorrect understanding then that is fine but you can just know he is wrong :)
------------------
So ... WHAT is the difference between these if they have exactly the same average damage?
1) Rolling 5 separate dice will do the same average damage over time but will do less damage more often.
2) There is only one circumstance where they will do different damage and that is when resistance is involved.
- 5 d8 rolled separately do 4/2 damage each for an average of 10 damage
- 5d8 rolled at once does 22.5/2 damage for an average of 11
So the only time these do different amounts of damage is when resistance is involved and even then the difference is very small and favors rolling all the dice at once.