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Independent damage dice versus doubling one face
C11 · Intermediate · Scoped component
Question and scope
A D&D 2014 critical hit rolls extra damage dice; an invented house rule doubles the damage number instead. For a d6 weapon, compare two independent d6 with twice one d6. Predict whether their means or possible results differ.
Source sections are linked under Rules below; primary-source review is recorded in the release dossier. Independent human and browser/usability review are separate gates.
Inputs and model limits
One fair d6 distribution reused independently in addition, versus multiplication of each possible face. Critical-damage example is conditional on a hit already being critical.
Complete runnable model
A roll variable stores a distribution, not a secretly sampled face. r + r combines two independent copies. r * 2 transforms each outcome of one roll by multiplication. Neither expression retrieves the faces of a previous runtime roll.
r = d(6)
output("Two independent damage dice", r + r)
output("House rule: double one face", r * 2)
output("Two dice can total seven", (r + r).pmf(7))
output("Doubled face can total seven", (r * 2).pmf(7))
Two independent damage dice · DieRoll · mean 7.000
| outcome | % | frac | X/36 |
|---|---|---|---|
| 2 | 2.78 | 1/36 | 1 |
| 3 | 5.56 | 1/18 | 2 |
| 4 | 8.33 | 1/12 | 3 |
| 5 | 11.1 | 1/9 | 4 |
| 6 | 13.9 | 5/36 | 5 |
| 7 | 16.7 | 1/6 | 6 |
| 8 | 13.9 | 5/36 | 5 |
| 9 | 11.1 | 1/9 | 4 |
| 10 | 8.33 | 1/12 | 3 |
| 11 | 5.56 | 1/18 | 2 |
| 12 | 2.78 | 1/36 | 1 |
House rule: double one face · DieRoll · mean 7.000
| outcome | % | frac | X/36 |
|---|---|---|---|
| 2 | 16.7 | 1/6 | 6 |
| 4 | 16.7 | 1/6 | 6 |
| 6 | 16.7 | 1/6 | 6 |
| 8 | 16.7 | 1/6 | 6 |
| 10 | 16.7 | 1/6 | 6 |
| 12 | 16.7 | 1/6 | 6 |
Two dice can total seven · Prob
| outcome | % | frac | X/36 |
|---|---|---|---|
| Two dice can total seven | 16.7 | 1/6 | 6 |
Doubled face can total seven · Prob
| outcome | % | frac | X/36 |
|---|---|---|---|
| Doubled face can total seven | 0.00 | 0 | 0 |
Reading the report and independent checks
Both means are 7 and ranges are 2–12. The sum can make every integer in between; doubled damage only makes even numbers. Seven has probability 1/6 for the sum and zero for the doubled face. Same mean does not imply same distribution.
Safe changes
Use d4 instead. Both means become 5; only the independent sum can total 5 (probability 1/4).
Composition and boundary checks
If a weapon adds +3 damage, which dice-only critical expression keeps that modifier once? Answer: r + r + 3, not (r + 3) * 2. Later joint callbacks preserve a shared face when a rule must inspect it more than once.
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
D&D Basic Rules 2014, Combat: Critical Hits. Only damage dice conditional on an ordinary critical hit; no hit chance or extra damage features. Doubling a face is explicitly a house rule.
Related learning
Lesson L18 explains the modelling idea step by step. Cookbook index offers both game and mechanic views of these same recipes.
Content ID: C11 · Review status: pilot
Prerequisites: L18