Concepts

Damage math for D&D, Baldur's Gate 3, Pathfinder, Diablo, Path of Exile, and anywhere else dice or weighted-RNG hits land — with a ground-up fundamentals track for readers who want the underlying probability machinery first. Each pillar explains one counterintuitive statistical fact most build guides skip and shows you the worked-out distributions, computed by an exact-rational engine you can drive yourself from the comparison tool.

Fundamentals

Probability distributions, expected value, variance, independence, and the normal approximation — explained on dice, in order. Start here if you want the language and the closed forms before the counterintuitive results.

What's a probability distribution?
From 1d6 (flat) to 2d6 (a tent) to 3d6 (smoother still). Sample space, PMF, support, and the convolution that takes you from one to the next.
Expected value, in dice
The weighted average. Linearity of expectation — E[X + Y] = E[X] + E[Y], always — and the (M+1)/2 shortcut for fair 1dM dice.
Variance and standard deviation
How spread-out a distribution is. The (M² − 1)/12 closed form for fair dice, and additivity of variance under independence.
Independence, sums, and convolutions
What independence means, why summing dice is convolving PMFs, when the closed-form mean and variance shortcuts apply, and when they don't.
When you can use a normal approximation
The central limit theorem on dice — visible in 1d6 → 2d6 → 4d6 → 8d6 — and four situations where the bell curve quietly lies.

Counterintuitive results

Each page below is one statistical fact that surprises most players the first time they see the worked-out probabilities. Every pillar links back to the relevant fundamentals as prerequisites.

Variance and kill probability
Why 2d6+5 beats 3d4+4 against an 11 HP target — even with nearly identical means. The single most counterintuitive thing in dice math.
When crit chance beats base damage
Two builds with identical expected DPR can have very different per-target kill rates. Where +1 damage wins and where +5% crit chance wins.
Reliable vs nuke builds
Multi-hit cleave vs single-big-swing. Same mean, opposite shapes. Which kills the target in fewer rounds depends on which side of the threshold you're on.
Advantage and disadvantage
Advantage isn't '+5 to hit on average.' It's a curve, peaking at AC ≈ to-hit + 11, falling to roughly +1 at the extremes. Worked out per-AC.
Exploding dice and fat tails
Why 1d6! is more than 1d6 + 0.7. The mean barely moves; the variance and the long right tail do all the work that makes the mechanic feel different at the table.
Expected strikes to kill
When HP grows past your single-strike maximum, the variance lesson flips: mean wins linearly and variance becomes a small constant correction. The elementary renewal theorem, applied to D&D.
Split damage and resistance
A flaming sword's 1d8 slash + 1d4 fire beats a 1d12 slash against a slash-resistant target — same dice budget, hedged across types. Per-component resistance math, worked out for D&D, BG3, and ARPGs.

Tactical questions

Specific verbatim questions players ask at the table. Each is built on the fundamentals above and applies one of the counterintuitive results to a particular feat, weapon, or encounter. Closed-form answers, no Monte Carlo.

Great Weapon Master break-even
When does -5/+10 stop being worth it? Closed-form rule: break-even AC ≈ attack-bonus + 8 for Greataxe-class weapons. Worked tables for D&D 2024.
Sharpshooter break-even
The ranged twin of GWM. Break-even AC ≈ attack-bonus + 7 for Longbow-class weapons. Plus the Crossbow Expert + Sharpshooter stack and the long-range disadvantage clause.
Great Weapon Fighting vs raw rerolls
The reroll-on-1-or-2 buys ~0.5–0.7 mean damage on common weapons — and reduces variance, which has knock-on effects on kill probability that the build guides skip.
Elven Accuracy
What does 3d20kh1 actually look like? Closed-form hit-chance curve, exact mean of 1981/120 ≈ 16.51, and the rule of thumb for whether to take it over the +2 ASI.
Finishing Strahd
The Curse of Strahd boss fight worked out. 144 HP vs typical level-10 party DPR. Why high-variance builds matter less here than at the table they feel.
Magic Missile vs Scorching Ray
When does auto-hit beat higher mean? Break-even AC ≈ 16 with +5 spell-attack mod. Plus the variance + resistance angles that keep MM winning more often than expected.