Polyhedral Dice Roller (D6, D20) & Coin Flipper | Cryptographic Random

CSPRNG Randomness & Elimination of Modulo Bias

In computer science and statistical modeling, generating fair random numbers across non-power-of-two ranges (such as a 6-sided or 20-sided die) introduces a well-documented vulnerability known as modulo bias. When a raw pseudo-random 32-bit unsigned integer (0 to 2³² - 1 = 4,294,967,295) is naively reduced via the modulo operator (X mod N), certain face outcomes occur with slightly higher frequency because 2³² is not evenly divisible by N.

To achieve uncompromising cryptographic fairness, this simulator implements rejection sampling:

limit = 2^32 - (2^32 mod sides)

Any random integer generated by crypto.getRandomValues() that equals or exceeds this threshold is rejected and re-sampled. This guarantees uniform probability down to the exact mathematical limit.

Tabletop RPG Dice Symbology & Probability Profiles

Die Type	Polyhedron Geometry	Discrete Uniform Probability	Primary TTRPG Mechanic (e.g. D&D 5e)
D4	Regular Tetrahedron (4 faces)	25.00% per face	Dagger damage, Magic Missile, Healing Word
D6	Regular Hexahedron / Cube (6 faces)	16.67% per face	Shortsword damage, Fireball damage, Rogue Sneak Attack
D8	Regular Octahedron (8 faces)	12.50% per face	Longsword damage, Cure Wounds, Cleric hit dice
D10	Pentagonal Trapezohedron (10 faces)	10.00% per face	Heavy crossbow, Halberd, Fighter hit dice
D12	Regular Dodecahedron (12 faces)	8.33% per face	Greataxe damage, Barbarian hit dice
D20	Regular Icosahedron (20 faces)	5.00% per face	Core D20 System: Attack rolls, ability checks, saving throws
D100	Zocchihedron / 2d10 Percentile	1.00% per integer	Wild Magic Surge table, Divine Intervention, Call of Cthulhu

Bernoulli Trials & The Law of Large Numbers

The flip of an unbiased coin is the archetypal Bernoulli trial: an experiment with exactly two mutually exclusive outcomes (Success p = 0.5, Failure q = 1 - p = 0.5). While short runs frequently exhibit localized clustering (e.g. four heads in five flips), the Strong Law of Large Numbers proves that as the sample size n → ∞, the sample mean X̄_n converges almost surely to the expected value μ = 0.5.

Frequently Asked Questions (US Standards)

How does this tool eliminate modulo bias in random number generation?

Standard implementations that use Math.floor(Math.random() * sides) or crypto.getRandomValues() % sides suffer from "modulo bias" when the range of the random integer generator is not an exact multiple of the desired sides. Our algorithm uses rejection sampling: it discards random integers that fall into the incomplete trailing remainder of the 32-bit unsigned integer range (4,294,967,296), guaranteeing mathematically pure, uniform probability distribution for every die face.

What polyhedral dice types are supported for tabletop RPGs?

The roller supports all seven standard polyhedral dice utilized in tabletop roleplaying games such as Dungeons & Dragons (D&D 5e), Pathfinder, and Call of Cthulhu: D4 (tetrahedral), D6 (cubic), D8 (octahedral), D10 (pentagonal trapezohedral), D12 (dodecahedral), D20 (icosahedral), and D100 (percentile).

How does the multi-coin batch simulation demonstrate the Law of Large Numbers?

A fair coin flip represents an idealized Bernoulli trial with a theoretical probability p = 0.5 for heads and p = 0.5 for tails. In small sample sets (e.g. 5 flips), variance and clustering are common. However, when flipping 100 coins in a batch, the Law of Large Numbers dictates that the empirical ratio of heads and tails will converge closely toward the expected 50.00% equilibrium.

Are roll or flip results transmitted over the network?

No. All entropy gathering, rejection sampling algorithms, and 3D CSS tumble animations execute 100% locally on your device within the browser JavaScript engine.