The core formula
For most attack rolls, expected damage per attack is the chance the attack hits multiplied by the average damage when it hits. Misses usually deal nothing, so they add zero to the sum.
Expected damage = hit chance × average damage on a hit
It is a direct use of the idea behind expected value, applied to one decision: which attack should I use this turn?
Step 1: Turn the roll into a hit chance
Take a d20 attack with a +5 bonus against a target number of 15. The attack hits when the d20 plus 5 reaches 15 or more, so you need to roll at least 10. The faces 10 through 20 are 11 faces out of 20, giving a hit chance of 11/20 = 0.55.
A quick way to do this: count how many faces succeed and divide by 20. Each +1 to the bonus adds one face, which is 5 percentage points.
Step 2: Find average damage
The average of a single die with n sides is (n + 1) / 2. A d8 averages 4.5 and a d6 averages 3.5. Add any flat bonus afterward. For several dice, average each and add.
- 1d8 + 3 averages 4.5 + 3 = 7.5
- 2d6 + 3 averages 7 + 3 = 10
- 1d6 + 3 averages 3.5 + 3 = 6.5
Step 3: Compare options
Say you can choose between a light weapon and a heavy one.
- Option A: +5 to hit, 1d8 + 3 damage. Hit chance 0.55. Expected damage 0.55 × 7.5 = 4.125.
- Option B: +3 to hit, 2d6 + 3 damage. You need a 12 or higher, which is 9 faces out of 20, so 0.45. Expected damage 0.45 × 10 = 4.5.
The heavier weapon is slightly ahead despite missing more often. The gap is small, about 0.4 damage per attack, so other considerations such as range, special effects or the target’s remaining health may reasonably decide the choice.
Adding critical hits
Many systems let a natural 20 deal extra damage, often by rolling the damage dice twice. The simplest way to handle this is to add the expected extra damage: the chance of a critical hit times the average of the extra dice.
With a 5% critical chance (one face in twenty):
- Option A: extra 1d8 averages 4.5, so the bonus is 0.05 × 4.5 = 0.225. New total: 4.125 + 0.225 = 4.35.
- Option B: extra 2d6 averages 7, so the bonus is 0.05 × 7 = 0.35. New total: 4.5 + 0.35 = 4.85.
This assumes a natural 20 always hits, which is already counted in both hit chances above. Rules differ between systems, so check yours before copying this step. The critical hits of video games follow the same logic, and they are covered in more depth in discussions of crit chance and damage output.
Several attacks in one turn
Expected values add, so multiple attacks are easy. Two attacks, each with a 0.55 hit chance and 1d6 + 3 damage, give 2 × 0.55 × 6.5 = 7.15 expected damage for the turn.
Compare that with a single stronger attack of 1d12 + 3 (average 9.5) at the same 0.55 hit chance: 0.55 × 9.5 = 5.225. On average the two smaller attacks win, but they also spread the result across two rolls, which makes a turn of zero damage less likely. With two attacks, the chance that both miss is 0.45 × 0.45 = 0.2025, about 20%. With one attack, it is 45%.
Advantage and disadvantage
Rolling two d20s and taking the higher result changes the hit chance. If a single roll hits with probability p, the chance that at least one of two rolls hits is 1 − (1 − p)². For p = 0.55 that is 1 − 0.45² = 1 − 0.2025 = 0.7975, roughly 80%. The attack from Option A then averages about 0.7975 × 7.5 ≈ 5.98 damage instead of 4.125, before critical hits. The effect of an advantage mechanic is largest when the base chance sits near the middle, and smaller when the base chance is already very high or very low.
What the average hides
Expected damage is a long-run figure. Over a single fight, a character can deal much more or much less than the number, particularly with only a few attacks. If an enemy has 6 hit points left and you need to finish it this turn, the better option may not be the one with the highest average; it is the one with the highest chance of dealing at least 6.
Expected damage also ignores many things players care about: movement, action economy, resistances, conditions, and the value of removing an enemy before it acts. Use it to compare options on equal footing, then adjust for context. Treat small differences, like those in the examples above, as ties and decide on other grounds.