[RPG] How to calculate average damage for DPR comparisons

damagednd-5estatistics

I have asked a few questions about optimization and DPR (damage per round) and I don't understand how those numbers are reached.

Take, for example, a monster with 15 AC and a level 1 fighter with a greatsword and +2 on strength.

I think that die average is just a simple divide by 2.

• d4 average = 2
• d6 average = 3
• d8 average = 4
• d10 average = 5
• d12 average = 6
• d20 average = 10

A greatsword does 2d6, so a average of 6 damage.

Adding prof and str to the roll makes gives it a + 4 making a d20 roll between 5 and 24.

Does this make a AC 15 with +4 the same as an AC 11 with +0?

An AC 11 gives 9 out of 20 chance to hit = 9/20 = 0.45 = 45% to hit.

Then the 6 damage divided by the percentile to hit for 0.45 * 6 = 2.7 damage per round.

Are my calculations correct or am I missing something?

i think that die average is just a simple divide by 2.

No, you need to add 0.5, since the damage is the average of all possible rolls. You could pair up the lowest and the highest, the second lowest and the second highest roll, ect. to get $$\\frac{sides}{2}\$$ pairs with sum $$\sides+1\$$ each. Divide the sum by the number of rolls in the pair (2) to get the average of $$\(sides+1) / 2\$$.

does this make a ac15 with +0 the same as an ac11 with +4?

That's the wrong way of expressing the fact you used, when calculating the number of positive outcomes. For each roll $$\X\$$ comparing $$\X+4\$$ with $$\AC\$$ and comparing $$\X\$$ with $$\AC-4\$$ yields the same result. Of course this means that the combinations of (AC 15, +4 attack bonus) and (AC 11, +0 attack bonus) have the same chance of hitting.

an AC 11 gives 9 out of 20 chance to hit = 9/20 = 0.45 = 45% to hit.

Not exactly. 11, 12, ..., 20 are $$\10 = (20 - (AC - modifier - 1)) = 21 + modifier - AC\$$ numbers.

are my calculations correct or am i missing something?

You're missing something (except for what's already mentioned): critical hits. Also you're not including your strength modifier in the damage calculation.

Correct calculation

The expected damage without considering the increased damage of a critical hit

$$E_{non-crit} = \frac{10}{20}\left(2\cdot\frac{1}{2}(6+1)+2\right) = \frac{90}{20} = 4.5$$

Now we also need to double the dice, if a critical hit happens (1/20 cases); note that the base damage and the damage dice of a normal hit were already included in the previous calculation:

$$E_{bonus\;crit} = \frac{1}{20} \cdot 2 \cdot \left(\frac{1}{2}(6+1)\right)= \frac{7}{20} = 0.35$$

Summing both expected damage values we get $$\E=4.5+0.35=4.85\$$ expected damage for a single attack.