Dice notation: NdN+M, advantage, and probability distributions

Apr 27, 2026 5 min read

“2d6+3”, “1d20”, “roll with advantage” — TTRPGs have their own dice notation. This article walks through how to read it and the probabilities.

NdN basics

AdB+C means “roll A dice of B sides each, add C”:

A — number of dice.
d — for “dice”.
B — sides per die.
+C — modifier (bonus).

Examples:

1d6 — one six-sided die (1–6).
2d6 — two d6 (2–12).
3d6+5 — three d6 plus 5 (8–23).
1d20 — one twenty-sided die (1–20).
2d10 — two d10 (2–20).

Common dice

Notation	Sides	Distribution	Typical use
d4	4	uniform 1–4	small weapons
d6	6	uniform 1–6	classic, used in many games
d8	8	uniform 1–8	medium weapons
d10	10	uniform 1–10	percentile base
d12	12	uniform 1–12	large weapons
d20	20	uniform 1–20	D&D core check
d100	100	uniform 1–100	percentile

A d100 is physically two d10s read as tens and ones.

Probability distributions

1d20 (uniform)

Each face has probability 1/20 = 5%.

2d6 (triangular)

Sum of two d6:

Sum	Combinations	Probability
2	(1,1)	1/36 = 2.78%
3	(1,2)(2,1)	2/36 = 5.56%
4	(1,3)(2,2)(3,1)	3/36 = 8.33%
5	4 ways	4/36 = 11.11%
6	5 ways	5/36 = 13.89%
7	6 ways	6/36 = 16.67%
8	5 ways	5/36
9	4 ways	4/36
10	3 ways	3/36
11	2 ways	2/36
12	(6,6)	1/36

7 is the most common — classic triangular distribution.

3d6 (more concentrated)

3d6 clusters around the mean (10–11). Different from 1d18+2:

3d6 — concentrated (std dev ~2.96).
1d18+2 — uniform (std dev ~5.19).

D&D ability scores (Strength etc.) traditionally come from 3d6 because extreme rolls are rare.

Advantage / disadvantage

Introduced in D&D 5e:

Advantage

Roll two d20s, take the higher.
Average rises 10.5 → 13.83 (~+3.3).
Probability of natural 20: 5% → 9.75%.

Disadvantage

Roll two d20s, take the lower.
Average drops 10.5 → 7.18 (~-3.3).
Probability of natural 1: 5% → 9.75%.

Often described as ”≈ +3 / -3 modifier”.

Exploding dice

Roll the max → roll again and add:

d6, roll a 6 → roll another d6 and add.
If that’s also a 6 → keep going.

Expected value of 1d6 rises from 3.5 to 4.2. Theoretically unbounded.

Keep / drop notation

4d6dl1-style:

dl1 — drop the lowest 1.
dh1 — drop the highest 1.
kh3 — keep the highest 3.
kl3 — keep the lowest 3.

D&D 5e ability score generation: “Roll 4d6, drop the lowest” = 4d6dl1.

Means:

4d6 — average 14.
4d6dl1 — average 12.24 (skewed higher).

Critical hits and fumbles

D&D 5e:

Natural 20 — automatic hit, damage dice doubled.
Natural 1 — automatic miss.

Each is 5% on a d20. With advantage 9.75%, with disadvantage 0.25%.

Call of Cthulhu (BRP)

A different system:

Roll 1d100.
Equal-or-under skill — success.
Half skill or less — “hard success”.
Fifth of skill or less — “extreme success / critical”.
96–100 — fumble.

Skill 60% — 60 or under succeeds (60% probability).

Implementing a roller

function rollDice(notation) {
	// parse "2d6+3"
	const match = notation.match(/(d+)d(d+)([+-]d+)?/);
	const count = parseInt(match[1]);
	const sides = parseInt(match[2]);
	const modifier = parseInt(match[3] || 0);

	let total = modifier;
	for (let i = 0; i < count; i++) {
		total += Math.floor(Math.random() * sides) + 1;
	}
	return total;
}

rollDice('2d6+3'); // 5..15

Caveats:

Math.random() is not cryptographically secure.
For online TRPG tools, sometimes crypto.getRandomValues() is used instead.

Mass dice

Many dice rolls approach the normal distribution (central limit theorem). 100d6:

Mean ≒ 350.
Std dev ≒ 17.08.
95% in 316–384.

Approximation suffices for “probability of 200d20 totaling 1000+“.

Extended notation

Notation	Meaning
`1d6+1d4`	mix d6 with d4
`1d6*10`	scale a d6 by 10
`2d20kh1`	advantage
`2d20kl1`	disadvantage
`4d6dl1`	drop the lowest of 4d6
`1d100` or `1d%`	percentile
`(1d6)d6`	roll 1d6 d6s

Summary

AdB+C = A dice of B sides plus modifier C.
1d20 is uniform; 2d6 triangular; 3d6 concentrated.
Advantage ≈ +3 modifier (take higher of two).
Exploding dice, keep/drop variants are common.
D&D uses d20; CoC uses 1d100.

To roll dice digitally, the dice roller on this site supports the common notations.