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An unbounded rule needs an explicit cap

Part G · L28 · Prerequisite: L27 · One modelling idea

Situation, question, and prediction

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.

Build

explode(d(6), max_depth=cap) allows at most cap extra rolls. max_depth=cap is a named argument: it supplies the value of our variable cap to the function’s max_depth option. 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

Acing d6: two extra rolls · DieRoll · mean 4.181

Terminal twelve at cap one · Prob

Run, read, and check

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.

Change one thing

Use cap zero: no extra roll is allowed and the mean returns to 3.5. Keep the output title honest when changing the cap.

Try it yourself

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.

Rules and model notes

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.

What you now know / where next

An unbounded rule needs an explicit cap is the reusable idea. Follow the generated previous/next links below, or return to the course index. For a complete self-contained application, see exploding-dice.

Content ID: L28 · Review status: pilot

Prerequisites: L27