Dice are among the oldest randomization devices in human history, with six-sided dice dating back over 5,000 years to ancient Mesopotamia. The mathematics of dice probability formed one of the earliest branches of formal probability theory, with Gerolamo Cardano's "Liber de Ludo Aleae" (Book on Games of Chance), written around 1564, being one of the first systematic treatments of probability, motivated entirely by dice gambling.

The probability distribution of a single fair die is uniform: each face has exactly a 1/n probability of appearing, where n is the number of faces. For the standard d6, each face has a 1/6 (approximately 16.67%) probability. When rolling multiple dice and summing the results, the distribution shifts from uniform to approximately normal (bell-shaped) due to the Central Limit Theorem. For example, rolling 2d6 produces a triangular distribution with values ranging from 2 to 12, where 7 is the most probable sum (probability 6/36 or about 16.7%) and 2 and 12 are the least probable (each 1/36 or about 2.8%). This non-uniform distribution is why game designers use 2d6 instead of 1d12 when they want results clustered around the middle: 2d6 produces 7 roughly six times more often than 2 or 12, while 1d12 gives each result an equal chance.

The polyhedral dice used in tabletop role-playing games (d4, d6, d8, d10, d12, d20) correspond to regular and semi-regular polyhedra. The d4 (tetrahedron), d6 (cube), d8 (octahedron), d12 (dodecahedron), and d20 (icosahedron) are the five Platonic solids, the only convex polyhedra where all faces are identical regular polygons. This geometric property is what makes them fair: the symmetry ensures each face has an equal probability of landing up. The d10, while not a Platonic solid, is a pentagonal trapezohedron with 10 congruent kite-shaped faces, designed to also be fair through bilateral symmetry.

Digital dice rollers must carefully consider their random number generation method. Pseudorandom number generators (PRNGs), like the Mersenne Twister used in many programming languages, produce sequences that are statistically random enough for gaming but are deterministic given their seed state. Cryptographically secure pseudorandom number generators (CSPRNGs), like those provided by the Web Crypto API's getRandomValues() function, use entropy from hardware sources (mouse movements, disk timing, thermal noise) to produce numbers that are computationally indistinguishable from true randomness. For dice rolling, either approach is more than adequate for fairness, but CSPRNGs provide the additional guarantee that results cannot be predicted even by someone who knows the algorithm.

The mathematics of fair dice extends beyond simple probability to the concept of Sicherman dice, discovered by Colonel George Sicherman in 1978. These are a pair of dice with non-standard numbering (1,2,2,3,3,4 and 1,3,4,5,6,8) that produce exactly the same sum distribution as standard 2d6. This remarkable result, proven through generating functions in algebra, demonstrates that the probability distribution of a sum depends on the mathematical structure of the dice, not just their physical appearance.