About

A miscounted roll or biased random generator can invalidate hours of tabletop gameplay. Standard Math.random implementations use pseudo-random number generators seeded from predictable entropy pools, producing sequences that fail statistical uniformity tests over large sample sizes. This tool generates each die face using the browser's crypto.getRandomValues API, which draws from OS-level entropy sources (e.g., /dev/urandom on Linux, CryptGenRandom on Windows). Rejection sampling eliminates modulo bias when mapping a uniform 0 - 255 byte range onto 6 outcomes. The tool computes running statistics - sum, mean, standard deviation - and maintains a full roll history for audit. It approximates ideal behavior assuming a perfectly fair die with equal probability 16 per face.

Formulas

Each die face X is a discrete uniform random variable over {1, 2, 3, 4, 5, 6} with probability mass function:

P(X = k) = 16 for k ∈ {1, 2, 3, 4, 5, 6}

The expected value for a single d6:

E[X] = 1 + 2 + 3 + 4 + 5 + 66 = 3.5

For n dice, the sum S = n∑i=1 X_i has expected value and variance:

E[S] = 3.5n Var(S) = 3512n σ = √35n12

Where n = number of dice rolled, X_i = result of the i-th die, S = total sum, σ = standard deviation. The variance of a single d6 is 3512 ≈ 2.917. By the Central Limit Theorem, for large n, S approximates a normal distribution N(3.5n, 35n12).

Reference Data

Number of Dice	Min Sum	Max Sum	Expected Value (μ)	Std. Deviation (σ)	Most Likely Sum	Common Use
1d6	1	6	3.500	1.708	Any (uniform)	Single checks, simple board games
2d6	2	12	7.000	2.415	7	Monopoly, Catan, Backgammon
3d6	3	18	10.500	2.958	10 - 11	D&D ability scores (straight)
4d6	4	24	14.000	3.416	14	D&D ability scores (drop lowest)
5d6	5	30	17.500	3.819	17 - 18	Yahtzee, risk pools
6d6	6	36	21.000	4.183	21	Fireball damage (D&D 5e)
8d6	8	48	28.000	4.830	28	High-level spell damage
10d6	10	60	35.000	5.401	35	Warhammer attack pools
12d6	12	72	42.000	5.916	42	Meteor Swarm (D&D 5e, fire portion)
15d6	15	90	52.500	6.614	52 - 53	Large Shadowrun dice pools
20d6	20	120	70.000	7.638	70	Mass combat, stress tests
36d6	36	216	126.000	10.247	126	Statistical demonstration
100d6	100	600	350.000	17.078	350	CLT demonstration, extreme pools

Frequently Asked Questions

The roller uses the Web Crypto API's crypto.getRandomValues(), which draws from OS-level entropy sources (hardware interrupts, thermal noise). Standard Math.random() uses algorithms like xorshift128+ that are deterministic given the seed. Additionally, this tool applies rejection sampling: a random byte (0-255) is generated, and values above 251 (the largest multiple of 6 below 256) are discarded and resampled. This eliminates modulo bias, where values 252-255 would make faces 1-4 slightly more probable than 5-6 (a bias of approximately 1.6%).

The "4d6 drop lowest" method rolls four six-sided dice and discards the single lowest result, summing the remaining three. This produces a distribution with expected value ≈ 12.24 (versus 10.5 for straight 3d6), skewing results toward higher ability scores. It is the standard method recommended in D&D 5th Edition (Player's Handbook, Chapter 1) for generating ability scores, as it reduces the probability of extremely low values: the chance of rolling a sum below 8 drops from ~9.3% (3d6) to ~1.6% (4d6 drop lowest).

The tool supports 1 to 100 dice per roll. The random number generation itself is near-instantaneous even at 100 dice since crypto.getRandomValues can fill a typed array of 100 bytes in microseconds. The visual animation renders up to 30 individual dice faces; for rolls above 30 dice, results are displayed as a summary list rather than individual animated dice to maintain 60fps rendering. Statistics (sum, mean, standard deviation) are computed in O(n) time, which is negligible for n ≤ 100.

The tool calculates the population standard deviation of the individual die results within a single roll. For a roll of n dice producing values x₁, x₂, …, xₙ, the mean μ = Σxᵢ/n is computed first, then σ = √(Σ(xᵢ − μ)²/n). This is the population formula (dividing by n, not n−1), since the roll represents the complete population of outcomes, not a sample from a larger dataset. The theoretical σ for a single d6 is √(35/12) ≈ 1.708.

The modifier is applied once to the total sum, not per die. If you roll 3d6 and set a modifier of +5, the tool computes Sum = (die₁ + die₂ + die₃) + 5. This matches the convention used in D&D and most tabletop RPGs where notation like "2d6+3" means roll two dice, sum them, then add 3. The individual die results displayed are unmodified; the modifier appears as a separate line item in the result breakdown.

Statistical convergence to the theoretical probability follows the law of large numbers, but convergence is slow. After n total die faces, the expected deviation from 1/6 for any single face is approximately 1/√(6n). For 100 rolls of 1d6, expect face frequencies to deviate by roughly ±4% from 16.67%. You would need approximately 10,000 individual die rolls to consistently see deviations below ±1%. The history panel's frequency distribution helps you track convergence over your session.