RPG Dice Probability

How are probabilities computed?

With XdY (X dice of Y faces), the tool walks through every possible combination (Y^X) and tallies how many land on each sum. A 3d6, for instance, gives 216 combinations.

Once the dice count gets large, say a 10d20, dynamic programming steps in instead of brute force and keeps the combinatorial explosion in check.

The computation is 100% local and works well for D&D, Pathfinder and other RPG systems.

Dice probability in role-playing games — fundamentals

Tabletop RPGs like Dungeons & Dragons, Pathfinder, Call of Cthulhu and GURPS are built on a single device: the polyhedral die. The classic seven-die set has the d4, d6, d8, d10, d00, d12 and the iconic d20. Each is meant to be a fair solid where every face has equal probability. Mathematically, a fair die follows a discrete uniform distribution: the chance of any face is exactly 1/N, where N is the number of faces.

For a single d20 the probability of any number from 1 to 20 is 1/20 = 5%. The mean of a uniform die with faces 1 to N is (N+1)/2, so a d20 averages 10.5, a d6 averages 3.5 and a d100 averages 50.5. Variance grows with range: a d20 has variance (20² − 1)/12 ≈ 33.25, much higher than a d6 (≈ 2.92). This is why d20 systems are famously "swingy" while sums of dice cluster tightly around the mean.

This calculator solves those equations exactly, without simulation. Enter a notation such as 2d6+3 and a target value; the tool enumerates every possible outcome to compute the exact probability of being equal, at least or at most that target. The bar chart visualises the full probability mass function.

Reading NdX+M notation: 1d20, 2d6+3, 4d6 drop lowest

The standard notation in modern RPGs is NdX±M: N dice with X faces, plus or minus a flat modifier M. Examples:

• 1d20 — a single twenty-sided die, used for attacks, saving throws and ability checks in D&D 5e.
• 2d6+3 — roll two six-sided dice, add them together, then add 3. Common damage roll for a +1 longsword.
• 3d6 — sum of three d6, used to generate the iconic 3-to-18 ability score in classic D&D and many OSR retroclones.
• 4d6 drop lowest — roll four d6, discard the smallest, sum the remaining three. The default ability score generation method in D&D 5e.
• 1d100 or d% — a percentile roll from 01 to 100, used for skill checks in Call of Cthulhu and Warhammer Fantasy Roleplay.

Modifiers shift the distribution by a constant: 2d6 goes from 2 to 12 with mean 7, so 2d6+3 goes from 5 to 15 with mean 10. The shape is unchanged — only the centre moves. A +5 modifier guarantees the worst 2d6+5 roll is still 7, well above a flat d20 median.

Uniform distribution vs sums of dice (bell curve)

By the central limit theorem, as you add more independent dice the distribution converges to a Gaussian "bell curve" — centred on the expected value with rapidly diminishing tails. Compare 2d6:

2d6 — 36 combinations
Sum:  2   3   4   5   6   7   8   9  10  11  12
Ways: 1   2   3   4   5   6   5   4   3   2   1
P(%): 2.78 5.56 8.33 11.11 13.89 16.67 13.89 11.11 8.33 5.56 2.78

The mode 7 has probability 6/36 ≈ 16.67%, six times more likely than either extreme (1/36 ≈ 2.78%). Now compare 3d6 (216 combinations, range 3-18):

3d6 — 216 combinations
Sum:    3    4    5    6    7    8    9   10   11   12   13   14   15   16   17   18
Ways:   1    3    6   10   15   21   25   27   27   25   21   15   10    6    3    1
P(%): 0.46 1.39 2.78 4.63 6.94 9.72 11.57 12.50 12.50 11.57 9.72 6.94 4.63 2.78 1.39 0.46

The central modes 10 and 11 each carry 12.50%, while the legendary 18 has only 1/216 ≈ 0.46% chance — the same as the minimum 3. For 4d6 drop lowest the mean rises to ≈ 12.24 and the chance of at least one 18 across six ability rolls climbs to ~9.3%, which is why 5e characters routinely have one stat near 18 but rarely two.

Advantage and disadvantage in D&D 5e

Advantage rolls 2d20 and keeps the higher; disadvantage keeps the lower. With X₁, X₂ independent uniform on {1,…,20}:

• P(advantage = y) = (2y − 1)/400
• P(disadvantage = y) = (2(21 − y) − 1)/400 = (41 − 2y)/400
• P(advantage ≥ y) = 1 − ((y − 1)/20)²
• P(disadvantage ≥ y) = ((21 − y)/20)²

Expected value with advantage is 13.825 and with disadvantage 7.175 — a 6.65-point swing on a d20, ≈ ±3.3 implicit modifier. The effect is non-linear: strongest on near-50/50 rolls, weaker at the extremes. Success table:

Target  Normal   Advantage   Disadvantage
5+      80.00%   96.00%       64.00%
10+     55.00%   79.75%       30.25%
15+     30.00%   51.00%        9.00%
18+     15.00%   27.75%        2.25%
20      5.00%    9.75%         0.25%

Advantage nearly doubles a 30% success rate (to 51%) and the natural 20 rate jumps from 5% to 9.75% — almost twice the crit rate. That dramatically affects crit-dependent builds (paladin smites, champion fighters, Hexblade's Curse).

System spotlight: D&D, Pathfinder, Call of Cthulhu, Storyteller

• D&D 5e / Pathfinder — d20 + modifier vs a DC. Flat uniform base; every +1 is +5% success. PF2 adds critical degrees (beat by 10 = crit, miss by 10 = fail).
• Call of Cthulhu (BRP) — d100 roll-under skill. Setting skill 5-99 sets the success rate directly. Hard halves skill; Extreme quarters; criticals on 01.
• World of Darkness — pool of d10, count successes (≥ 8). Chance of at least one success in N dice is 1 − (7/10)^N. 5 dice = 83.2%; 10 dice = 97.2%.
• GURPS — 3d6 roll-under, so the 3d6 bell curve shapes odds directly. Skill 12 = 74.07%; skill 14 = 90.74%.
• Powered by the Apocalypse — 2d6 + stat: 6− failure, 7-9 mixed, 10+ full. The 2d6 curve makes the mixed band (≈ 41.67% at +0) the most common outcome by design.

Applications: encounter balance, optimal play, character builds

Knowing the probabilities lets a GM tune encounters by maths. A level-5 party with +7 to hit against AC 15 succeeds at (21 − 15 + 7)/20 = 65%. Three attacks per round give 1.95 expected hits — enough to drop a 30 HP minion in two rounds. Shift +to-hit or AC by 2 and the kill rate moves 10-15%.

The same maths drives optimal play: Bless (+1d4, mean +2.5) is multiplicative with Sneak Attack, lifting hit rate by ~12.5% on rolls that fire 6d6 damage. Choosing between Faerie Fire (advantage) and Hex (+1d6)? At hit rates 40-60% Faerie Fire wins because that is exactly where the advantage curve has the biggest delta.

FAQ

Why doesn't this tool simulate rolls? Simulation has 1-2% noise. This tool enumerates the full sample space, so probabilities are exact rationals.

What does 4d6 drop lowest do to the mean? Raises it from 10.5 (3d6) to ≈ 12.24 and trims the low tail. A result of 3 drops from 1/216 to 1/1296.

How rare is a natural 20? Flat d20: 5%. Advantage: 1 − (19/20)² = 9.75%. Disadvantage: (1/20)² = 0.25% — once every 400 rolls.

Is 3d6 really less swingy than d20? Yes — σ(3d6) ≈ 2.96 vs σ(d20) ≈ 5.77. Results within ±1σ cover ~68% of 3d6 outcomes but only ~58% of d20.

Does Elven Accuracy double advantage? It rolls 3d20 and keeps the highest: P(≥ y) = 1 − ((y − 1)/20)³. Against DC 15, 65.7% (vs 51% advantage); the natural 20 hits 14.26% of the time.