Dice notation step by step

At the table

Tabletop games write rolls as XdY (X dice, Y sides), sometimes with a bonus or drop lowest. This lesson walks from the smallest form to 4d6dl1 ability scores.

Try it

In the playground, open this lesson in the playground and click Run (or press Shift+Enter). Output lists five distributions in order—one per stage below.

Stage 1: one die (`1d4`)

↗output("one_d4", 1d4)

one_d4 · DieRoll · mean 2.500

outcome	%	frac	X/36
1	25.0	1/4	9
2	25.0	1/4	9
3	25.0	1/4	9
4	25.0	1/4	9

two_d6 · 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

two_d6_plus_3 · DieRoll · mean 10.000

outcome	%	frac	X/36
5	2.78	1/36	1
6	5.56	1/18	2
7	8.33	1/12	3
8	11.1	1/9	4
9	13.9	5/36	5
10	16.7	1/6	6
11	13.9	5/36	5
12	11.1	1/9	4
13	8.33	1/12	3
14	5.56	1/18	2
15	2.78	1/36	1

four_d6 · DieRoll · mean 14.000

outcome	%	frac	X/36
4	0.08	1/1295	0
5	0.31	1/324	0
6	0.77	5/648	0
7	1.54	5/324	1
8	2.70	34/1259	1
9	4.32	7/162	2
10	6.17	5/81	2
11	8.02	13/162	3
12	9.65	106/1099	3
13	10.8	35/324	4
14	11.3	73/648	4
15	10.8	35/324	4
16	9.65	106/1099	3
17	8.02	13/162	3
18	6.17	5/81	2
19	4.32	7/162	2
20	2.70	34/1259	1
21	1.54	5/324	1
22	0.77	5/648	0
23	0.31	1/324	0
24	0.08	1/1295	0

four_d6dl1 · DieRoll · mean 12.245

outcome	%	frac	X/36
3	0.08	1/1295	0
4	0.31	1/324	0
5	0.77	5/648	0
6	1.62	7/432	1
7	2.93	19/648	1
8	4.78	31/648	2
9	7.02	62/883	3
10	9.41	61/648	3
11	11.4	37/324	4
12	12.9	167/1296	5
13	13.3	43/324	5
14	12.3	10/81	4
15	10.1	103/1019	4
16	7.25	47/648	3
17	4.17	1/24	2
18	1.62	7/432	1

four_d6dh1 · DieRoll · mean 8.755

outcome	%	frac	X/36
3	1.62	7/432	1
4	4.17	1/24	2
5	7.25	47/648	3
6	10.1	103/1019	4
7	12.3	10/81	4
8	13.3	43/324	5
9	12.9	167/1296	5
10	11.4	37/324	4
11	9.41	61/648	3
12	7.02	62/883	3
13	4.78	31/648	2
14	2.93	19/648	1
15	1.62	7/432	1
16	0.77	5/648	0
17	0.31	1/324	0
18	0.08	1/1295	0

four_d6kh2 · DieRoll · mean 9.344

outcome	%	frac	X/36
2	0.08	1/1295	0
3	0.31	1/324	0
4	1.16	5/432	0
5	2.47	2/81	1
6	5.02	49/977	2
7	8.33	1/12	3
8	13.2	19/144	5
9	17.3	14/81	6
10	20.1	29/144	7
11	18.8	61/324	7
12	13.2	19/144	5

three_d12kl1 · DieRoll · mean 3.521

outcome	%	frac	X/36
1	23.0	190/827	8
2	19.2	263/1373	7
3	15.7	93/593	6
4	12.6	163/1298	5
5	9.78	129/1319	4
6	7.35	94/1279	3
7	5.27	89/1690	2
8	3.53	55/1558	1
9	2.14	27/1261	1
10	1.10	17/1546	0
11	0.41	5/1234	0
12	0.06	1/1726	0

2d6 — roll two six-sided dice and add them.
Sum of two independent d6 (same idea as lesson 2); the engine expands this to dice_pool(2, 6) and then sums. For pools you keep separate, see lesson 6.

Mean ≈ 7.

Stage 3: flat bonus (`2d6 + 3`)

↗output("two_d6_plus_3", 2d6 + 3)

4d6 — roll four d6 and sum all of them (no drops).

Mean ≈ 14 (four times the mean of one d6).

Stage 5: drop lowest (`4d6dl1`)

↗output("four_d6dl1", 4d6dl1)

dh1 — drop highest 1 die, then sum the rest (the opposite of dl1).
Expands to drop_highest(4, 6, 1).

Mean ≈ 8.76.

Stage 7: keep highest N (`4d6kh2`)

↗output("four_d6kh2", 4d6kh2)

kl1 — keep the lowest 1 die and sum it (here, just that single die).
Expands to keep_lowest(3, 12, 1).

Mean ≈ 3.52 (average of the minimum of three d12).

Suffix cheat sheet

Suffix	Meaning	Function
`dlN`	Drop lowest N, sum the rest	`drop_lowest`
`dhN`	Drop highest N, sum the rest	`drop_highest`
`khN`	Keep highest N, sum those	`keep_highest`
`klN`	Keep lowest N, sum those	`keep_lowest`

Reading the result

Eight named outputs in the sample script, each with mean and a probability table (or truncated table for large supports).

Try this

Change 1d4 to a die your game uses (1d8, 1d12).
Compare 2d6 + 3 and shift(2d6, 3)—means should match.
Try 4d6kh3 vs 4d6dl1—both sum three dice, but different sets of faces count.