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Keeping three equals dropping one

Part D · L13 · Prerequisite: L12 · One modelling idea

Situation, question, and prediction

D&D 2014 offers random ability generation by rolling four d6 and adding the highest three. Compare one such score with 3d6. Predict whether dropping one positive die can increase the sum of those same four dice.

Build

4d6dl1 means four d6, drop the lowest one; 4d6kh3 means keep the highest three. Plain notation such as roll = 4d6 preserves a pool, so use roll.sum() for its total. Keep/drop notation such as score = 4d6dl1 already returns a numeric distribution: do not call .sum() on that score. The function form keep_highest(4, 6, 3) also returns the kept total.

score = keep_highest(4, 6, 3)
output("One 4d6 drop-lowest score", score)
output("Three d6", dice_pool(3, 6).sum())
output("All four d6", dice_pool(4, 6).sum())

One 4d6 drop-lowest score · DieRoll · mean 12.245

outcome	%	frac	X/1296
3	0.08	1/1295	1
4	0.31	1/324	4
5	0.77	5/648	10
6	1.62	7/432	21
7	2.93	19/648	38
8	4.78	31/648	62
9	7.02	62/883	91
10	9.41	61/648	122
11	11.4	37/324	148
12	12.9	167/1296	167
13	13.3	43/324	172
14	12.3	10/81	160
15	10.1	103/1019	131
16	7.25	47/648	94
17	4.17	1/24	54
18	1.62	7/432	21

Three d6 · DieRoll · mean 10.500

outcome	%	frac	X/1296
3	0.46	1/216	6
4	1.39	1/72	18
5	2.78	1/36	36
6	4.63	5/108	60
7	6.94	5/72	90
8	9.72	7/72	126
9	11.6	25/216	150
10	12.5	1/8	162
11	12.5	1/8	162
12	11.6	25/216	150
13	9.72	7/72	126
14	6.94	5/72	90
15	4.63	5/108	60
16	2.78	1/36	36
17	1.39	1/72	18
18	0.46	1/216	6

All four d6 · DieRoll · mean 14.000

outcome	%	frac	X/1296
4	0.08	1/1295	1
5	0.31	1/324	4
6	0.77	5/648	10
7	1.54	5/324	20
8	2.70	34/1259	35
9	4.32	7/162	56
10	6.17	5/81	80
11	8.02	13/162	104
12	9.65	106/1099	125
13	10.8	35/324	140
14	11.3	73/648	146
15	10.8	35/324	140
16	9.65	106/1099	125
17	8.02	13/162	104
18	6.17	5/81	80
19	4.32	7/162	56
20	2.70	34/1259	35
21	1.54	5/324	20
22	0.77	5/648	10
23	0.31	1/324	4
24	0.08	1/1295	1

Run, read, and check

The score ranges 3–18 and has mean 12.2445987654. That exceeds 3d6’s 10.5, but is below 4d6’s 14. Discarding a positive die cannot increase that same throw’s total. Keeping three and dropping one have identical complete distributions, not just equal means.

Change one thing

Keep the highest two instead. This equals dropping the lowest two; its mean is 9.34413580247 and range 2–12.

Try it yourself

Explain why six independent score reports are not a complete character-generation policy. Answer: choosing assignments, reroll rules, and the quality of the whole array need additional questions. This page models one score only.

Rules and model notes

D&D Basic Rules 2014, Step-by-Step Characters, Determine Ability Scores. One random score; no reroll policy, point buy, or six-score array ranking.

What you now know / where next

Keeping three equals dropping one is the reusable idea. Follow the generated previous/next links below, or return to the course index. For a complete self-contained application, see ability-scores-4d6dl1.

Content ID: L13 · Review status: pilot

Prerequisites: L12