Dice probability explained: why 7 is the most common roll, and how to work out any odds

A pair of dice is the oldest probability lesson there is, and most of what the casino and board-game industries know comes from counting its 36 outcomes. This guide does the counting, answers the questions people actually ask, and explains what the dice roller is doing when it simulates a fair die. Is a coin flip really 50/50? covers the two-sided version and the fallacies.

One die

A fair six-sided die gives each face a probability of 1/6 ≈ 16.7%, every roll, regardless of history. The average roll is 3.5 (the mean of 1 to 6), which no single roll can show but which a hundred rolls approach. The chance of not rolling a six is 5/6; of not rolling one in n rolls, (5/6)ⁿ — so the chance of at least one six in four rolls is 1 − (5/6)⁴ ≈ 51.8%, the bet the Chevalier de Méré made money on in the 1650s before losing it on the next one.

Two dice: the triangle

Two dice give 6 × 6 = 36 equally likely ordered outcomes. Sums are not equally likely because different numbers of outcomes produce them: 2 in one way (1+1), 3 in two, 4 in three, up to 7 in six ways (1+6, 2+5, 3+4, 4+3, 5+2, 6+1), then down to 12 in one. So P(7) = 6/36 = 1/6, P(2) = P(12) = 1/36, and the distribution is a triangle centred on 7 — the shape that makes 6 and 8 the "point" numbers in craps and 7 the number that ends the round. Doubles: 6 of 36, or 1/6. A specific double (two sixes): 1/36.

The common questions

• At least one six on two dice: 1 − (5/6)² = 11/36 ≈ 30.6% — not 1/3, because the double six is counted once.
• Sum of 7 or 11 (a natural): 8/36 = 22.2%.
• Both dice the same as a target (e.g. two 4s): 1/36.
• Higher roll wins: ties 6/36; each player wins 15/36.
• Rolling a 6 before a 7 (craps point of 6): 5/11 ≈ 45.5% — only the outcomes that end the sequence matter.
• Advantage (roll two, keep higher, as in some role-playing games): average rises from 3.5 to 4.47; the chance of at least a 5 goes from 33% to 56%.

The percentage calculator converts the fractions; Fractions, decimals and percentages: the same number three ways, and how to convert explains the conversions.

Many dice and the bell curve

Add more dice and the sum's distribution goes from a triangle toward a bell: three dice (3–18) peak at 10 and 11 with 12.5% each, and the extremes (3 or 18) are 1/216 each. This is the central limit theorem in miniature — sums of independent random things tend to a normal distribution — and it is why role-playing games use 3d6 for "average" traits (most characters near the middle) and a single d20 for swingy outcomes (every result equally likely). The mean of n dice is 3.5n; the spread grows only with √n, so the total becomes more predictable in relative terms as dice are added (Standard deviation explained: what it measures, how it's calculated, and the 68–95–99.7 rule).

Fair dice and loaded ones

Real dice are slightly unfair: mass-produced dice with rounded corners and drilled pips have a tiny bias, casino dice are machined to a tenth of a millimetre and have flush pips to avoid it. Testing fairness is a counting exercise: roll many times and compare each face's frequency to 1/6 — a chi-squared test tells you whether the deviation is beyond chance, and it takes hundreds of rolls to detect a small bias. Software dice, including the roller here, use the operating system's random number generator; How random is a random number generator? covers how random that is and why it's random enough for games but a different question for cryptography. The random number generator rolls any-sided dice; Yacht puts five of them to work.