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How much does an explosion cap matter?
C09 · Advanced · Bounded approximation
Question and scope
A generic acing damage component rolls again on a d6 maximum and adds the result. Real unlimited chains have no finite upper bound. With one extra roll allowed, predict why a total of 12 appears even though unlimited acing would continue.
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Inputs and model limits
One fair d6. Integer extra-roll cap 0–6; default 1, compared with 2. No unlimited-distribution claim.
Complete runnable model
explode(d(6), max_depth=cap) allows at most cap extra rolls. The terminal roll counts even when it shows 6. Compare caps rather than calling the bounded result an exact model of an infinite rule.
cap = 1
bounded = explode(d(6), max_depth=cap)
output("Acing d6: one extra roll", bounded)
output("Acing d6: two extra rolls", explode(d(6), max_depth=2))
output("Terminal twelve at cap one", bounded.pmf(12))
Acing d6: one extra roll · DieRoll · mean 4.083
| outcome | % | frac | X/36 |
|---|---|---|---|
| 1 | 16.7 | 1/6 | 6 |
| 2 | 16.7 | 1/6 | 6 |
| 3 | 16.7 | 1/6 | 6 |
| 4 | 16.7 | 1/6 | 6 |
| 5 | 16.7 | 1/6 | 6 |
| 7 | 2.78 | 1/36 | 1 |
| 8 | 2.78 | 1/36 | 1 |
| 9 | 2.78 | 1/36 | 1 |
| 10 | 2.78 | 1/36 | 1 |
| 11 | 2.78 | 1/36 | 1 |
| 12 | 2.78 | 1/36 | 1 |
Acing d6: two extra rolls · DieRoll · mean 4.181
| outcome | % | frac | X/36 |
|---|---|---|---|
| 1 | 16.7 | 1/6 | 6 |
| 2 | 16.7 | 1/6 | 6 |
| 3 | 16.7 | 1/6 | 6 |
| 4 | 16.7 | 1/6 | 6 |
| 5 | 16.7 | 1/6 | 6 |
| 7 | 2.78 | 1/36 | 1 |
| 8 | 2.78 | 1/36 | 1 |
| 9 | 2.78 | 1/36 | 1 |
| 10 | 2.78 | 1/36 | 1 |
| 11 | 2.78 | 1/36 | 1 |
| 13 | 0.46 | 1/216 | 0 |
| 14 | 0.46 | 1/216 | 0 |
| 15 | 0.46 | 1/216 | 0 |
| 16 | 0.46 | 1/216 | 0 |
| 17 | 0.46 | 1/216 | 0 |
| 18 | 0.46 | 1/216 | 0 |
Terminal twelve at cap one · Prob
| outcome | % | frac | X/36 |
|---|---|---|---|
| Terminal twelve at cap one | 2.78 | 1/36 | 1 |
Reading the report and independent checks
At cap one, faces 1–5 each have probability 1/6 and totals 7–12 each 1/36; no 6 is possible. Mean is 49/12. At cap two, mean is 301/72. The unlimited mean is 4.2, so the first cap’s mean error is 7/60, not 1/36.
Safe changes
Use cap zero: no extra roll is allowed and the mean returns to 3.5. Keep the output title honest when changing the cap.
Composition and boundary checks
What is the probability the cap interrupts a continuing d6 chain? Answer: (1/6)^(cap+1), hence 1/36 at cap one. Under a shared roll sequence that bounds disagreement on an event, not mean damage error. Checkpoint: distinguish bounded explosion, once-only reroll, and conditioning.
Do not combine separate marginal reports and assume they preserve the same rolled faces. When an event depends on several properties of one roll, keep those properties together inside a single classifier or pool callback.
Rules
Generic bounded acing component. Not a complete Savage Worlds trait roll: no Wild Die, critical failure, modifiers, raises, or Bennies. A pool of exploding dice needs careful performance analysis; exploding a sum is a different rule.
Related learning
Lesson L28 explains the modelling idea step by step. Cookbook index offers both game and mechanic views of these same recipes.
Content ID: C09 · Review status: pilot
Prerequisites: L28