About

Tabletop RPG outcomes hinge on the d20 roll. A miscounted modifier or a forgotten advantage condition can invalidate an entire combat round. This calculator uses crypto.getRandomValues() with rejection sampling to produce uniform distribution across 1 - 20, eliminating modulo bias that plagues naive Math.random() implementations. It supports multi-die pools up to 100d20, signed modifiers from −999 to +999, and D&D 5e Advantage/Disadvantage mechanics. Roll history persists across sessions for audit trails during long campaigns.

The expected value of a single d20 is 10.5. For n dice with modifier m, expected total equals n × 10.5 + m. Advantage shifts the single-die expected value to approximately 13.825. Disadvantage drops it to roughly 7.175. This tool computes these statistics in real time. Note: this tool approximates a perfectly fair icosahedron. Physical dice with manufacturing defects will deviate from theoretical distributions.

Formulas

The total result of a dice roll with n dice and modifier m is computed as:

Total = n∑i=1 r_i + m

where each r_i ∈ {1, 2, …, 20} is drawn from a uniform discrete distribution.

Expected value for a single d20:

E[d20] = 1 + 202 = 10.5

Standard deviation for n dice:

σ = √n × 20² − 112 ≈ 5.766 × √n

For Advantage (rolling 2d20, keeping the highest):

E[Adv] = 2 × 20 + 13 = 13.833

For Disadvantage (keeping the lowest):

E[Disadv] = 20 + 23 ≈ 7.167

Rejection sampling ensures uniform distribution: a random byte from crypto.getRandomValues() in range 0 - 255 is accepted only if < 240 (largest multiple of 20), then mapped via r = (byte mod 20) + 1.

Where: n = number of dice, m = modifier (signed integer), r_i = individual die result, σ = standard deviation.

Reference Data

Roll Mode	Dice Expression	Min	Max	Expected Value	Std Dev (σ)	P(Natural 20)	P(Natural 1)	Common Use
Standard	1d20	1	20	10.50	5.77	5.00%	5.00%	Attack rolls, ability checks
Advantage	2d20 keep highest	1	20	13.83	4.72	9.75%	0.25%	Flanking, Reckless Attack
Disadvantage	2d20 keep lowest	1	20	7.18	4.72	0.25%	9.75%	Dodge, Poisoned condition
Standard	2d20	2	40	21.00	8.16	N/A (sum)	N/A (sum)	Dual rolls, group checks
Standard	1d20+5	6	25	15.50	5.77	5.00%	5.00%	Level 5 fighter attack
Standard	1d20−2	−1	18	8.50	5.77	5.00%	5.00%	Low ability modifier
Standard	3d20	3	60	31.50	9.99	N/A (sum)	N/A (sum)	Multi-attack totals
Standard	4d20	4	80	42.00	11.55	N/A (sum)	N/A (sum)	Mass roll pools
Advantage	2d20kh+3	4	23	16.83	4.72	9.75%	0.25%	Advantage + modifier
Disadvantage	2d20kl+3	4	23	10.18	4.72	0.25%	9.75%	Disadvantage + modifier
Standard	1d20 (DC 15)	1	20	10.50	5.77	30% pass	70% fail	Medium difficulty check
Standard	1d20 (DC 10)	1	20	10.50	5.77	55% pass	45% fail	Easy difficulty check
Standard	1d20 (DC 20)	1	20	10.50	5.77	5% pass	95% fail	Hard difficulty check
Advantage	Adv (DC 15)	1	20	13.83	4.72	51% pass	49% fail	Adv on medium DC
Disadvantage	Disadv (DC 15)	1	20	7.18	4.72	9% pass	91% fail	Disadv on medium DC

Frequently Asked Questions

A random byte ranges from 0 to 255, giving 256 possible values. Since 256 is not evenly divisible by 20, naive modulo (byte % 20) would make values 0-15 slightly more probable than 16-19. Rejection sampling discards bytes ≥ 240 (the largest multiple of 20 within 256), ensuring each of the 20 outcomes has exactly a 1/20 probability. The rejection rate is only 6.25%, so performance impact is negligible.

Advantage rolls two d20s and keeps the maximum. The probability of getting exactly value k as the maximum is P(max = k) = (2k − 1) / 400. This weights higher values exponentially more than a single roll. The expected value computes to (2 × 20 + 1) / 3 ≈ 13.833. This represents roughly a +3.3 effective bonus compared to a straight roll, though the actual benefit varies by target DC.

Per D&D 5e rules (PHB p.173), multiple sources of Advantage do not stack - you still roll only 2d20 and take the highest. If you have both Advantage and Disadvantage from any number of sources, they cancel out regardless of quantity, and you roll a single d20 (straight roll). This calculator follows these official rules when both toggles are active.

For a straight d20 roll with modifier m against DC target t, the probability of success is P(success) = (21 − t + m) / 20, clamped between 0 and 1. A natural 20 always hits and a natural 1 always misses on attack rolls (but not ability checks per RAW). For DC 15 with +0 modifier, P = 6/20 = 30%. With Advantage, P(success) = 1 − ((t − m − 1) / 20)², giving approximately 51% for the same DC 15.

crypto.getRandomValues() uses a cryptographically secure pseudorandom number generator (CSPRNG) seeded from OS entropy sources (hardware interrupts, thermal noise). With rejection sampling applied, the output passes all NIST statistical randomness tests. Physical dice, by contrast, suffer from manufacturing imperfections: rounded edges, paint weight differences, and air bubble inclusions can create measurable bias. A 2009 study by Purdue University found off-the-shelf d20s can deviate up to 3% from expected frequencies. The digital implementation is statistically more fair.

Yes. If the modifier m is sufficiently negative, the total can fall below zero. For example, 1d20 − 5 yields a range of −4 to 15. In D&D 5e, negative ability check or attack roll totals are valid results and simply fail against any positive DC or AC. Damage rolls cannot go below 0 per RAW (PHB p.196). This calculator displays negative totals accurately without clamping, leaving interpretation to the game system in use.