Critical hit math: when crit chance lies to you

A 25% crit chance does not mean one in every four hits crits, and a bigger crit number does not always mean more damage. Here's the math that tells you what a crit stat is actually worth.

Published: 2026-08-21By: Toolhub Admin · About the author

Two builds sit in front of you. One adds crit chance, one adds crit damage, and the character sheet shows a "crit stat" going up in both cases. Which one actually does more damage? And separately: you have 25% crit chance, you attack four times, and you crit zero times — is the game cheating you, or is that just normal? Both questions have clean, checkable answers, and both are routinely botched by players who trust the number on the tooltip instead of the math underneath it.

The core problem is that a crit chance is a probability, and probabilities lie to intuition in two specific ways: they collapse into an average that hides how wildly individual results swing, and they tempt you into believing past results change future ones. This article unpacks both, then turns the math into a table you can use to compare builds.

The one formula that matters: expected value

A critical hit deals your base damage multiplied by some crit multiplier — commonly 2.0× (double), but games vary. On a normal hit you deal base damage; on a crit you deal base × critMult. Over many hits, the average damage per hit is:

avg damage = base × (1 + critChance × (critMult − 1))

The term critChance × (critMult − 1) is the bonus a crit adds on top of a normal hit, weighted by how often it happens. With a 25% chance and a 2.0× multiplier: 0.25 × (2.0 − 1) = 0.25, so your average hit is 1.25 × base — a 25% damage increase. Notice it is only the extra damage (the "−1") that gets scaled by chance, because you were always going to deal the base amount.

This single expected-value multiplier is the honest way to compare crit builds. It also settles the "more crit chance vs more crit damage" argument immediately: the two stats are symmetric inside the product. Whether you raise critChance or raise (critMult − 1), the marginal value depends on which is currently smaller. If your crit chance is already high and your multiplier is low, buy multiplier; if your multiplier is high and your chance is low, buy chance. The one that is lagging is worth more per point.

Why 25% crit "feels" unreliable: variance

Expected value tells you the long-run average. It says nothing about how bumpy the ride is, and the bumpiness is what you actually feel while playing. Each attack is a Bernoulli trial — it crits or it doesn't — and a run of attacks follows a binomial distribution. For a 25% crit chance over four attacks, the chance of getting zero crits is (0.75)^4 ≈ 0.316. Nearly one time in three, four attacks produce no crit at all. That is not the game cheating; that is exactly what 25% means.

The spread is the problem. Two builds can share the same average damage while feeling completely different:

• A high-chance, low-multiplier build (say 60% chance, 1.5×) crits often, so results cluster tightly around the average. It feels steady and predictable.
• A low-chance, high-multiplier build (say 20% chance, 3.5×) delivers the same average but in rare, huge spikes. Long dry stretches punctuated by a monster hit. Higher variance, "swingier" feel.

Which you prefer is a real decision, not just flavour. For content where a single burst can end a fight, high variance is good — you only need one big roll. For sustained encounters where consistency keeps you alive, low variance is safer. The average alone cannot tell you this; you have to think about the distribution.

The gambler's fallacy: streaks owe you nothing

Here is the trap that costs players the most mental energy. You have missed four crits in a row. Surely the next one is "due"? In a true independent-trials system, no. Each attack rolls fresh. A 25% chance is 25% on the very next swing regardless of the entire history before it. The dice have no memory. Believing otherwise — that a run of failures makes success more likely, or that a hot streak must "regress" on the next roll — is the gambler's fallacy, and it is one of the most robust errors in human probabilistic reasoning.

What is true is that long streaks become rare as the sample grows: over hundreds of attacks your observed crit rate converges toward 25%. But that convergence happens because later results dilute the streak, not because the game "pays back" a debt. There is no debt. Nine tails in a row still leaves the tenth flip at 50/50.

Except when the game bends the rules: pseudo-random distribution

Some games deliberately break independence to make crits feel fairer, because true randomness produces streaks that players read as bugs. The best-known technique is pseudo-random distribution (PRD), popularised by the Warcraft III and Dota engines. Instead of rolling a flat probability every time, PRD starts each attack with a small chance that increases after every non-crit and resets after a crit. A displayed "25%" is implemented as a lower starting probability that ramps up, tuned so the long-run average still lands at 25%.

The effect: back-to-back crits become less likely, and very long droughts become almost impossible. The variance shrinks. A related idea is the "pity" or "soft/hard pity" system in gacha and loot games, where a guaranteed reward triggers after a set number of failures — the extreme end of the same principle. The takeaways for a player:

• In an independent system, the gambler's fallacy is a fallacy: streaks mean nothing about the next roll.
• In a PRD or pity system, a drought genuinely does raise your next-hit odds — the "due" feeling is, unusually, correct. But only in those systems, and only because the designer engineered it.
• You usually cannot tell which system you are in from the tooltip. When in doubt, assume independence; it is the safer default for planning.

Crit, attack speed, and multi-hit

Crit is a per-hit multiplier, which is why it composes cleanly with anything that changes how many hits you land. Because DPS is roughly avgDamagePerHit × hitsPerSecond, a percentage crit gain and a percentage attack-speed gain multiply, not add. A build that is +25% from crit and +20% from attack speed is 1.25 × 1.20 = 1.50, a 50% DPS increase, not 45%. This multiplicative stacking is why crit and speed are such a strong pairing.

Multi-hit abilities interact more subtly. If each sub-hit of a three-hit strike rolls its own crit independently, your average is unchanged — expected value is linear, so three small hits average the same as one big hit of equal total base. What changes is variance: three independent rolls smooth the outcome, so multi-hit skills feel more consistent than a single big swing with the same crit stats. If instead the ability rolls crit once and applies it to all sub-hits, you get the opposite — an all-or-nothing spike with much higher variance. The tooltip rarely says which; the feel of the ability usually tells you.

A worked DPS table

Effective DPS gain from crit is just the EV multiplier minus one: critChance × (critMult − 1). Below, several common chance/multiplier combos, assuming independent per-hit rolls and no other modifiers.

Two rows stand out. A 25%/3.0× build and a 50%/2.0× build both land at +50% average — identical EV, very different variance. And 50%/3.0× matches 100%/2.0× at +100% average, but the guaranteed-crit build has zero variance while the half-chance build swings hard. Same expected damage, different experience.

Tool walkthrough

Working the numbers by hand gets tedious fast, which is where two of Toolhub's utilities help. The damage calculator takes base damage, crit chance, and crit multiplier and reports the expected-value multiplier directly, so you can pit two builds against each other and see which EV wins before committing gear or points. Because the tool exposes the underlying 1 + critChance × (critMult − 1) term, it also makes the "which lagging stat is worth more" question concrete — nudge each input and watch the marginal gain.

To build intuition for variance rather than averages, the dice roller lets you simulate independent trials at a chosen probability and watch real streaks appear. Roll a batch at 25% and count how often four rolls in a row come up empty — you will see the ~32% dry-streak rate from earlier show up in practice. Seeing the clumping first-hand is the fastest cure for the gambler's fallacy: the streaks are normal, and the next roll never remembers them.

Where to read further

• Wikipedia: Expected value — the formal basis for the EV multiplier and why averages are linear across independent hits.
• Wikipedia: Gambler's fallacy — the reasoning error behind "my crit is due," and why independent trials have no memory.
• Wikipedia: Binomial distribution — the model behind streak probabilities like the (0.75)^4 dry-run calculation.

The tooltip gives you a chance and a multiplier; the math gives you what they are actually worth. Compare builds on expected value, choose your variance deliberately based on the content you are running, and stop reading meaning into streaks unless you have real evidence the game runs a pity or pseudo-random system. Do that, and the crit number stops lying to you.