What a critical hit chance really is
A critical hit is an extra roll. After an attack lands, the game checks whether it was a critical, and if so applies a damage multiplier. The critical hit chance is simply the probability that this check succeeds on a given attack. It says nothing about the next attack: each check is independent unless the game states otherwise.
Players often collect several sources of crit chance at once, such as a skill, an item and a temporary buff. The question is how those sources combine. There are two common designs, and the answer depends on which one a game uses.
Design one: bonuses are added together
Many games keep a single crit chance stat and simply add bonuses to it. A base 5% plus an item giving 10% plus a buff giving 15% yields 30%. Here the stacking is plain addition because the game sums everything before making one roll.
The usual catch is a cap. Chance cannot exceed 100%, and some games set a lower limit. Past the cap, extra bonuses do nothing for that stat. If a build adds up to 110% and the cap is 100%, the last 10% was wasted.
Design two: each source rolls separately
In other designs, each source makes its own independent roll. A critical happens if any one of them succeeds. These chances do not add. To combine them, it is easier to ask the opposite question: what is the chance that every source fails?
This is the complement rule. For independent sources, multiply the chances of failing, then subtract from one.
Take two sources with 20% and 30% chances (an illustration, not any particular game):
- Chance the first fails: 0.8
- Chance the second fails: 0.7
- Chance both fail: 0.8 × 0.7 = 0.56
- Chance at least one succeeds: 1 − 0.56 = 0.44
The combined chance is 44%, not 50%. The shortfall exists because in the cases where both sources would succeed, you only get one critical, not two.
Why the gap grows with more sources
Now take three independent sources of 10% each:
- Chance all three fail: 0.9 × 0.9 × 0.9 = 0.729
- Chance at least one succeeds: 1 − 0.729 = 0.271
A plain sum would suggest 30%, but the real figure is 27.1%. Each additional source adds a little less than the one before it, and the total can never reach 100% no matter how many are stacked. This is a general feature of combining independent chances, and it is the same reasoning behind the mistake of adding probabilities wrongly.
The gap is small when chances are small and grows as they get larger. Two 50% sources give 1 − 0.5 × 0.5 = 75%, not 100%.
Chance is only half of the story: crit damage
A crit chance matters through its effect on damage. Suppose a critical hit multiplies damage by m, and the crit chance is c. A normal hit deals 1 unit. The average damage per hit is then
(1 − c) × 1 + c × m = 1 + c × (m − 1)
With c = 0.44 and m = 2, the average is 1 + 0.44 × 1 = 1.44 times normal damage. With c = 0.25 and m = 1.5, it is 1 + 0.25 × 0.5 = 1.125.
This is a simple use of expected value. It also shows that crit chance and crit multiplier work together: raising either increases the average, but the benefit of extra chance depends on how large the multiplier is. When m is only slightly above 1, even a very high crit chance changes little.
Average versus actual results
An average is not a promise. With a 44% crit chance, a short fight can easily produce far fewer crits than expected, or far more. Over a few swings, luck dominates. Over thousands, the results tend to settle near the average, as described by the law of large numbers, but that does not help in any one fight.
This is also why a streak of crits does not mean the game is rigged, and a dry run does not mean one is “due”. Past attacks do not change the next roll, so believing otherwise is the gambler’s fallacy.
Practical checks when reading a game
When you want to know how bonuses combine, a few questions help:
- Does the game add bonuses into one stat, or does each source roll on its own?
- Is there a cap on the final chance?
- Do bonuses multiply each other rather than add? Some games treat percentage bonuses as multipliers on the base value, which changes the arithmetic again.
- Does the game use a pseudo-random distribution that adjusts odds after misses? If so, the stated percentage describes the long-run rate rather than each roll.
Game wording is often ambiguous, and tooltips can be simplified, so confirm against the game’s own documentation or testing where possible. The key habit is to separate the chance of an event from its payoff, and to combine independent chances through the chance of none of them happening.