Concepts

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.

The weighted average. Linearity of expectation — E[X + Y] = E[X] + E[Y], always — and the (M+1)/2 shortcut for fair 1dM dice.

How spread-out a distribution is. The (M² − 1)/12 closed form for fair dice, and additivity of variance under independence.

What independence means, why summing dice is convolving PMFs, when the closed-form mean and variance shortcuts apply, and when they don't.

The central limit theorem on dice — visible in 1d6 → 2d6 → 4d6 → 8d6 — and four situations where the bell curve quietly lies.

Why 2d6+5 beats 3d4+4 against an 11 HP target — even with nearly identical means. The single most counterintuitive thing in dice math.

Two builds with identical expected DPR can have very different per-target kill rates. Where +1 damage wins and where +5% crit chance wins.

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 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.

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.

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.

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.

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.

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.

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.

What does 3d20kh1 actually look like? Closed-form hit-chance curve, exact mean of 1239/80 ≈ 15.49, and the rule of thumb for whether to take it over the +2 ASI.

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.

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.

Fireball halves the total damage on save, not each die. DC matters more than slot level — and 3+ targets is where AOE wins back the slot cost. Save-or-suck contrast for the binary cases.

Smite always adds expected damage on a hit; the real choice is which hits get the slot. Smite-only-on-crit delivers exactly 1.875× more damage per slot than smite-first-hit — a constant that holds across slot levels and base-damage profiles.

Companion to the GWM break-even pillar. The kill-trigger half of GWM's bonus action — invisible against bosses (+0.16 DPR), dominant against minion queues (+15-40 DPR depending on shape, 28× the boss-fight number for PAM/GWM L5 vs four 12-HP minions). The cascade is a bounded Markov chain.

The four-way per-shot outcome split with exact rationals. Why graze is the underweighted half (a 65% hit / 15% graze shot deals damage 80% of the time, not 65%), how P(crit) is conditional on hit, and what differs from D&D's d20 attack model. Companion to /games/xcom/.

AGE-system stunts fire on success AND any-pair-matches. The product-law estimate P(stunt) = P(success) · P(doubles) is exactly right at TN=11 with no modifiers — a coincidence of 3d6's mirror symmetry — and wrong everywhere else. E[SP | stunt] is 217/48 ≈ 4.52, well above the uniform stunt-die mean of 3.5.

BRP percentile checks split a single d100 into six bands — critical, extreme, hard, regular, fail, fumble — with closed-form ratios that hold across every canonical CoC skill. Headline counterintuitive result: fumble probability jumps from 2/100 at S=49 to 5/100 at S=50, the only spot where investing in a skill makes catastrophic failure more likely.

Bonus / penalty die isn't kh1 / kl1 on a d100 — only the tens digit gets the order statistic. The advantage curve peaks sharply at skill 50 (a 25-point swing in P(regular)) and tapers to 9 points at S=10 or S=90. Bonus dice favour mid-skill characters; specialists past S=80 get less leverage.

Sanity is mathematically just HP with per-encounter damage and slow trickle healing. The same engine call that says "this monster will kill you in N strikes" says "this campaign will break your investigator in M sessions". Cure ratio decides whether the chain converges; variance shapes the survivors curve.