Concepts → Exploding dice and fat tails
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Why exploding dice are more than "+0.7 to your average roll"
A 1d6 averages 3.5; an exploding 1d6 averages 4.2. The mean shift is modest. The shape change is enormous, and that shape is what makes the mechanic feel different at the table, not the +0.7 mean.
Answer: exploding 1d6 mean is exactly
21/5 = 4.2 — only +0.7 over plain 1d6. The fat tail
is what carries the table feel: P(≥7) jumps from 0 to ~14%,
P(≥10) from 0 to ~3%.
The mechanic, briefly
On a maximum-face roll, roll the die again and add. If the second
roll also maxes, roll a third. And so on, in principle without
bound. Notation: 1d6!, 1d8!, etc.
Real-world implementations: Savage Worlds (every die "aces" on max), Fudge / FATE (variant rules), exploding-crit homebrew variants of 5e where a nat-20 damage die rolls another. The diceplots engine truncates at depth 20; the truncation error is ~6⁻²⁰ ≈ 3×10⁻¹⁶ for d6, well below floating-point noise.
The mean shift is small, but predictable
Closed form for the expected value of an exploding M-sided die:
E[dM!] = M(M+1) / (2(M−1))
| Die | Normal mean | Exploding mean | Increase |
|---|---|---|---|
| d4 | 2.5 | 10/3 ≈ 3.33 | +33% |
| d6 | 3.5 | 21/5 = 4.20 | +20% |
| d8 | 4.5 | 36/7 ≈ 5.14 | +14% |
| d10 | 5.5 | 55/9 ≈ 6.11 | +11% |
| d12 | 6.5 | 78/11 ≈ 7.09 | +9% |
| d20 | 10.5 | 210/19 ≈ 11.05 | +5% |
Smaller dice get bigger relative boosts, because the chance of exploding (1/M) is larger. A d4 explodes 25% of the time; a d20 only 5%.
The shape change is huge
Side by side: a flat 1d6 has six equally-likely faces, mean 3.5, max 6. An exploding 1d6 has the same flat distribution from 1–5, zero probability at exactly 6 (you'd reroll), then a small bump at 7–11, a smaller bump at 13–17, an even smaller bump at 19–23, and so on out to absurdly high values (with vanishingly small probability).
That bumpy fat tail is the whole story. Probabilities at high values:
| Outcome ≥ | P(1d6) | P(1d6!) |
|---|---|---|
| 6 | 1/6 ≈ 16.7% | 1/6 ≈ 16.7% |
| 7 | 0% | 1/6 ≈ 16.7% |
| 10 | 0% | 3/36 ≈ 8.3% |
| 13 | 0% | 1/36 ≈ 2.8% |
| 20 | 0% | ~0.13% |
| 30 | 0% | ~0.0021% |
A flat d6 caps at 6. An exploding d6 has a real (if small) chance to hit 13, 20, 30, or higher. In any system where one swing can threshold a kill, that fat tail is where exploding dice earn their reputation.
Variance amplification — biggest on small dice
With X = M·K + Y where K is the geometric "number of explosions" and Y the final non-max roll, both independent:
Var(dM!) = M³ / (M−1)² + M(M−2) / 12
For d6 that's 216/25 + 2 = 10.64: about
3.65× the variance of a flat d6 (which is
35/12 ≈ 2.92). For d20: 8000/361 + 30 = 52.16 vs
flat 33.25, only 57% more. The explodingness
of the mechanic shrinks as M grows. A d20! adds a lot less tail
relative to its flat baseline than a d6! does.
The intuition for why d6! feels so swingy at the table is now purely a variance argument: the same thing that drives the variance-vs-mean crossover. Fat tails make exploding dice extra-good when the target HP sits well above your normal-roll maximum, and slightly bad when you'd one-shot it on a flat roll anyway.
Where this matters in practice
Savage Worlds. Every Trait roll potentially aces. The system is built around the fat tail; "Wild Cards" additionally roll a parallel Wild Die that also aces, doubling down on the right-tail mass.
Homebrew exploding-crit 5e. House rule where a nat 20 on a damage die rolls again. Inflates damage output most against high-AC, high-HP bosses, where you crit infrequently but each crit can punch above the normal-roll ceiling.
Card-based combat (Slay the Spire-likes). Many card effects "deal damage equal to X plus rolling-bonus" with a fat right tail. Same statistical profile.
Anywhere a "lucky proc" mechanic exists. Diablo's "Lucky Hit" effects, Path of Exile's "trigger on reaching X damage". Fat-tailed distributions decide how often the trigger fires meaningfully.
Try it yourself
↦ /strike/1d6! — full exploding distribution ↦ /strike/1d6 — flat baseline ↦ /vs/1d8!/1d10 — exploding d8 vs flat d10 (similar means)