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.

When session prep calls for “how often does 2d6+3 hit X with advantage?” the fastest answer is to roll many times and look at the distribution. The dice roller on this site supports multiple notations at once, which makes it easy to spot-check encounter math before the game. Physical dice are still better for the actual game — this is for pre-session sanity checks, not table play.