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D&D 2014 — Fireball damage given the save result

R04 · Intermediate · Scoped component

Question and scope

D&D 2014 Fireball at its base spell level deals 8d6 damage, halved on a successful Dexterity save. Given that the save succeeded, what is mean damage? Predict whether it is exactly half of 28.

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

Base spell level, 8d6; one target. Reports are conditional on success/failure, not weighted by a target’s save chance.

Complete runnable model

dice_pool(8, 6).sum() efficiently builds the ordinary total. // 2 divides each possible integer result and rounds down. This is a transformation of the entire distribution, not division of a summary number.

full = dice_pool(8, 6).sum()
half = full // 2
output("Damage given failed save", full)
output("Damage given successful save", half)

Damage given failed save · DieRoll · mean 28.000

outcome	%	frac	X
8	0.00	0	0.000
9	0.00	0.000	0.000
10	0.00	0.000	0.000
11	0.01	0.000	0.000
12	0.02	1/5064	0.000
13	0.05	1/2117	0.000
14	0.10	1/983	0.001
15	0.20	2/997	0.002
16	0.37	4/1093	0.004
17	0.62	4/641	0.006
18	1.00	13/1299	0.010
19	1.52	10/659	0.015
20	2.18	23/1053	0.022
21	2.99	5/167	0.030
22	3.92	44/1123	0.039
23	4.90	49/999	0.049
24	5.88	424/7207	0.059
25	6.77	84/1241	0.068
26	7.48	115/1538	0.075
27	7.94	128/1613	0.079
28	8.09	48/593	0.081
29	7.94	128/1613	0.079
30	7.48	115/1538	0.075
31	6.77	84/1241	0.068
32	5.88	424/7207	0.059
33	4.90	49/999	0.049
34	3.92	44/1123	0.039
35	2.99	5/167	0.030
36	2.18	23/1053	0.022
37	1.52	10/659	0.015
38	1.00	13/1299	0.010
39	0.62	4/641	0.006
40	0.37	4/1093	0.004
41	0.20	2/997	0.002
42	0.10	1/983	0.001
43	0.05	1/2117	0.000
44	0.02	1/5064	0.000
45	0.01	0.000	0.000
46	0.00	0.000	0.000
47	0.00	0.000	0.000
48	0.00	0	0.000

Damage given successful save · DieRoll · mean 13.750

outcome	%	frac	X
4	0.00	0.000	0.000
5	0.01	0.000	0.000
6	0.07	1/1495	0.001
7	0.30	1/331	0.003
8	0.99	37/3738	0.010
9	2.52	31/1231	0.025
10	5.18	45/869	0.052
11	8.82	395/4477	0.088
12	12.7	125/988	0.127
13	15.4	84/545	0.154
14	16.0	193/1204	0.160
15	14.2	51/358	0.142
16	10.8	115/1066	0.108
17	6.91	77/1114	0.069
18	3.70	69/1864	0.037
19	1.62	20/1231	0.016
20	0.57	2/353	0.006
21	0.15	1/672	0.001
22	0.03	1/3719	0.000
23	0.00	0.000	0.000
24	0.00	0	0.000

Reading the report and independent checks

Full mean is 28; successful-save mean is 13.75, not 14. Half the possible probability mass is on odd totals, where rounding loses half a point, giving a mean loss of 1/4. Half-damage range is 4–24.

Safe changes

Use one d6 instead: full mean 3.5, rounded-half mean 1.5. Verify by writing transformed faces 0,1,1,2,2,3.

Composition and boundary checks

Why is this not the overall expected damage to a target? Answer: it conditions on the save result. You must also know the target’s save chance; L32 builds that full scoped question. Checkpoint: keep the conditioning statement in every output label.

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, Fireball and the basic round-down convention. Base third-level spell, one target; no resistance, vulnerability, Evasion, or other damage reductions.

Related learning

Lesson L20 explains the modelling idea step by step. Cookbook index offers both game and mechanic views of these same recipes.

Content ID: R04 · Review status: pilot

Prerequisites: L20