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Reusing a distribution is not reusing a face

Part E · L18 · Prerequisite: L17 · One modelling idea

Situation, question, and prediction

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.

Build

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

Run, read, and check

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.

Change one thing

Use d4 instead. Both means become 5; only the independent sum can total 5 (probability 1/4).

Try it yourself

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.

Rules and model notes

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.

What you now know / where next

Reusing a distribution is not reusing a face is the reusable idea. Follow the generated previous/next links below, or return to the course index. For a complete self-contained application, see independent-versus-shared.

Content ID: L18 · Review status: pilot

Prerequisites: L17