The Basic Formula
A critical hit is an attack that, with some probability, deals extra damage. Two numbers define it: the crit chance, written here as c (a value between 0 and 1), and the crit multiplier, written as m. A normal hit deals base damage B. A critical hit deals B times m.
The average damage per hit is the sum of each outcome weighted by its probability:
average = (1 - c) x B + c x B x m
Factoring out B gives a simpler form:
average = B x (1 + c x (m - 1))
The term c x (m - 1) is the average bonus from crits. This is an application of expected value: you weight each possible result by how likely it is and add them up.
A Worked Example
Take a base damage of 50, a 20% crit chance (c = 0.2) and a crit multiplier of 2 (m = 2).
- Normal hits happen 80% of the time and deal 50.
- Crits happen 20% of the time and deal 100.
- Average = 0.8 x 50 + 0.2 x 100 = 40 + 20 = 60.
Using the shortcut, 50 x (1 + 0.2 x 1) = 50 x 1.2 = 60. Both routes agree, and the crits have raised average output by 20%.
That average is not the damage of any single hit. Every individual hit deals either 50 or 100. The 60 is what you would approach over many hits, which is the idea behind the law of large numbers, and a short fight can land well above or below it.
Chance Versus Multiplier
Players often ask which stat is worth more. The formula shows that they feed into the same bonus term, c x (m - 1), so the answer depends on current values.
- Adding to crit chance increases the bonus by (m - 1) for each point of chance. With m = 2 each extra 1 percentage point of chance adds 0.01 to the multiplier on average damage.
- Adding to the multiplier increases the bonus by c for each unit added. With c = 0.2, raising m from 2 to 2.25 adds 0.2 x 0.25 = 0.05.
- In the same situation, raising c from 0.2 to 0.25 adds 0.05 x 1 = 0.05. The two upgrades are equal here.
When the multiplier is high, chance becomes more valuable. When chance is already high, the multiplier matters more. This also explains why the same percentage bonus is worth different amounts to different builds, and why chance has a hard ceiling: it cannot exceed 100%, at which point only the multiplier can improve output.
Different Builds, Same Average
Consider two builds with identical base damage of 50 and the same average bonus:
- Build A: 20% chance, multiplier 2, bonus 0.2 x 1 = 0.2, average 60.
- Build B: 10% chance, multiplier 3, bonus 0.1 x 2 = 0.2, average 60.
The averages match, but the experience differs. Build B has a lower chance of any crit but each one is bigger. This is where variance enters. For Build A, hits are 50 with probability 0.8 and 100 with probability 0.2. The variance per hit is 0.8 x (50 - 60)^2 + 0.2 x (100 - 60)^2 = 0.8 x 100 + 0.2 x 1600 = 80 + 320 = 400, so the standard deviation is 20. For Build B, hits are 50 with probability 0.9 and 150 with probability 0.1. The variance is 0.9 x 100 + 0.1 x 8100 = 90 + 810 = 900, with a standard deviation of 30.
Build B has the same average but a wider spread, so its results swing more from fight to fight. The general relationship is covered in game variance, and it matters for choosing builds: a high-spread build can be satisfying when it works but less reliable, especially in short encounters.
Where the Simple Formula Stops Working
The formula assumes several things that do not always hold.
- Hits must land. If an attack can miss, multiply by the hit probability. Expected output per attack is the hit chance times the average damage per hit.
- Crits may not be independent. Some games use systems that adjust the odds after a miss or a hit, so the rolls are not truly independent events. Pseudo-random distribution is one such system, and its long-run average still matches the stated rate even though short streaks are altered.
- Damage is often a range. If base damage varies, use its average for B, provided the crit multiplier applies to the whole result.
- Additive and multiplicative bonuses differ. Some games add the crit bonus to other bonuses instead of multiplying, which changes the shortcut. Check how the game stacks them before relying on a formula.
- Defense and caps change the picture. Armor that subtracts a flat amount before or after the crit multiplier changes what a crit is worth.
Tabletop players use the same idea when calculating expected damage in tabletop combat, with dice replacing the hidden roll.
Practical Takeaways
For a quick comparison, convert each option to the multiplier on average damage, 1 + c x (m - 1), and compare. Do not forget that a long fight reduces the influence of luck while a short fight leaves it large. If two options are close, consider the one with the smaller spread for reliability or the larger spread for a chance at a standout fight. Finally, keep in mind that the average is a guide, not a forecast: in a single encounter, crits arrive in clumps and gaps, and that does not mean the stated chance is wrong.