Dice Probabilities, Part 1: The Basics

A fair six-colour die gives each face a probability of 1/6, or about 16.67%. That is the whole model. Everything else in this article is arithmetic built on top of it.

Rolls are independent: the die has no memory. The chance of red twice in a row is 1/6 × 1/6 = 1/36, roughly 2.8%. Three reds is 1/216. But the chance of red on your next roll, given ten reds already, is still 1/6. Believing otherwise is the gambler's fallacy, and it is the single most expensive mistake in games of chance.

With multiple dice the useful question changes from 'which face' to 'how many'. Rolling n dice, the number showing red follows a binomial distribution: the expected count is n/6, and the probability of getting exactly k reds is C(n,k) × (1/6)^k × (5/6)^(n-k).

Two quick landmarks. With 6 dice, the chance of at least one red is 1 − (5/6)^6 ≈ 66.5%. With 12 dice it is about 88.8%, and with 25 dice about 99.0%. Notice that 'at least one' climbs fast while 'exactly one' peaks early and then falls — a distinction that trips up a lot of house-rule designers.

Streaks look wrong because humans expect randomness to alternate. In 100 rolls of a six-sided die, a run of four identical faces somewhere in the sequence is more likely than not. If your roller never produced streaks, that would be the evidence of tampering.

Part 2 covers the harder cases: matching sets, at-least-k thresholds, and how to size a dice pool so a target outcome lands at a chosen frequency.