Calculating damage increase from attack bonuses

[Saiga]

I've read before how the Archery fighting style is considered quite powerful and how +1 to hit is better than +1 to damage - however, is there a way to reliably calculate how much better things are? To me, +2 to hit for Archery is just a 10% increase on a d20 roll, which is less of an improvement over +2 to damage from Duelist (25% increase on a d8). I had thought a 10% increase in accuracy would just be a 10% increase in damage, but I guess that is too simplified.

Can anyone help me out with the math here? Is there a relatively simple way to calculate the value of any bonuses to attack rolls?

[Aett_Thorn]

I don't think it's simple, because it depends on the AC of the target.

For instance, if, without the bonus, you needed to roll a 20 to hit the target, then with the bonus, you need to roll 18+. Thus, you are looking at doing three times the damage.

However, if you only need to roll a 10 to hit without the bonus, now you need an 8. This is an expected damage increase of just 10%.

So it changes based on how difficult the target is to hit.

[mgshamster]

H*(D+B) + C*D

H = Chance to hit an AC on a d20
D = Damage Dice
B = Damage bonus/modifier
C = Chance to crit

Chance to hit example:
You have a +5 and need to hit an AC of 18. That's a 13 on the d20. Counting from 13-20, that's 8 possible results on the d20 that would count as a hit, so you have 8*0.05 = 0.4 chance to hit.

[Balyano]

Can anyone help me out with the math here? Is there a relatively simple way to calculate the value of any bonuses to attack rolls?

Take your % chance to hit after the +2 and divide it by your % chance to hit before the +2.

First column is the roll you need on a D20 to hit before adding the +2 to hit.
Second column is the percent chance to hit before the +2.
Third column is the percent chance to hit after the +2.
Fourth column is the damage multiplier for having the +2 over normal.

NOTE: THIS DOES NOT FACTOR IN CRITICAL HIT DAMAGE.

d20 - % - +2 - increase
1 - 100 - 100 - x1
2 - 95 - 100 - x1.0526315789473684210526315789474
3 - 90 - 100 - x1.1111111111111111111111111111111
4 - 85 - 95 - x1.1176470588235294117647058823529
5 - 80 - 90 - x1.125
6 - 75 - 85 - x1.1333333333333333333333333333333
7 - 70 - 80 - x1.1428571428571428571428571428571
8 - 65 - 75 - x1.1538461538461538461538461538462
9 - 60 - 70 - x1.1666666666666666666666666666667
10 - 55 - 65 - x1.1818181818181818181818181818182
11 - 50 - 60 - x1.2
12 - 45 - 55 - x1.2222222222222222222222222222222
13 - 40 - 50 - x1.25
14 - 35 - 45 - x1.2857142857142857142857142857143
15 - 30 - 40 - x1.3333333333333333333333333333333
16 - 25 - 35 - x1.4
17 - 20 - 30 - x1.5
18 - 15 - 25 - x1.6666666666666666666666666666667
19 - 10 - 20 - x2
20 - 5 - 15 - x3

[DivisibleByZero]

H*(D+B) + C*D

H = Chance to hit an AC on a d20
D = Damage Dice
B = Damage bonus/modifier
C = Chance to crit

Chance to hit example:
You have a +5 and need to hit an AC of 18. That's a 13 on the d20. Counting from 13-20, that's 8 possible results on the d20 that would count as a hit, so you have 8*0.05 = 0.4 chance to hit.

Slight amendment:
That's 7 possible results for normal damage, and one result for crit damage.
Total damages from those two, each multiplied by their own respective chance-to-hit, and add them together.

Example:
Level 1 warlock with 16 Cha using EB vs. AC 14:
+5 attack for 1d10 (average 5.5) damage
50% chance for normal hit: 5.5*0.5=2.75
5% chance to crit: 11*0.05=0.55
average damage per round calculating chance to hit: 2.75+0.55= ~3.3 vs. AC14

That same warlock, using Hex:
+5 attack for 1d10+1d6 (average 9) damage
50% chance for normal hit: 9*0.5=4.5
5% chance to crit: 18*0.05=0.9
average damage per round calculating chance to hit: 4.5+0.9= ~5.4 vs. AC14

That same warlock, at level 2 with agonizing blast:
+5 attack for 1d10+3 (average 8.5) damage
50% chance for normal hit: 8.5*0.5=4.25
5% chance to crit: 17*0.05=0.85
average damage per round calculating chance to hit: 4.25+0.85= ~5.1 vs. AC14

That same warlock, using Hex:
+5 attack for 1d10+1d6+3 (average 12) damage
50% chance for normal hit: 12*0.5=6
5% chance to crit: 24*0.05=1.2
average damage per round calculating chance to hit: 6+1.2= ~7.2 vs. AC14

To answer the OP, your individual damage will be higher with a damage boost, but your overall damage will be higher with an attack boost. Damage boost = slightly bigger numbers less consistently. Attack boost = slightly lower number more consistently, which adds up to more over time.
If you want to see it in action, use the format I showed and compare the exact same character build, using dex for both, a rapier and Dueling for one, and a longbow with Archery for the other. You'll see that the Duelist does bigger numbers when he hits, but that the Archer does more damage overall. It takes a little while for this to become true, but it does indeed become true after some levels.
It's quantity vs quality.
Unless the quality numbers are vastly superior, quantity wins over time.
At lower levels, that +2 damage is significant enough to make it superior unless the enemy AC is really high.
In the mid-levels, the attack bonus is usually better unless the enemy AC is terrible.
In the higher levels, it's about even, with extreme ACs tending to one side or the other depending on which extreme we're looking at.

Heck, it's slow at work, I'll do it for you.
Take a level 8 Ranger with Hunter's Mark and Dex 20 vs AC 18

Rapier with Dueling:
+8 attack for 1d8+1d6+7 (average 15) damage
45% chance for normal hit from attack: 15*0.45= ~6.75
5% chance to crit: 30+0.05= ~1.5
45% chance for normal hit from extra attack: 15*0.45= ~6.75
5% chance to crit: 30+0.05= ~1.5
average damage per round calculating chance to hit: 6.75+6.75+1.5+1.5= ~16.5 vs. AC18

Longbow with Archery
+10 attack for 1d8+1d6+5 (average13) damage
55% chance for normal hit from attack: 13*0.55= ~7.15
5% chance to crit: 26+0.05= ~1.3
55% chance for normal hit from extra attack: 13*0.55= ~7.15
5% chance to crit: 26+0.05= ~1.3
average damage per round calculating chance to hit: 7.15+7.15+1.3+1.3= ~16.9 vs. AC18

As the enemy AC climbs, so does the Archer's advantage.

[mgshamster]

Slight amendment:
That's 7 possible results for normal damage, and one result for crit damage.
Total damages from those two, each multiplied by their own respective chance-to-hit, and add them together.

You make a good point to consider. My equation works with the assumption that the damage dice for a normal attack will be the same as a crit.

So if you roll 1d8 normally, and it's doubled on a crit, my equation accounts for both dice. Once in the normal section, and once in the crit section. As soon as the crit dice become different than the normal attack dice, my equation doesn't work and your caveat becomes more accurate.

Mine was just a Q&D. :)

[Saiga]

Thanks for the input everyone! I definitely understand the value of it now, and how to properly look at it.