Dice probability

Dice probabilities are counted, not estimated. This tool builds the exact distribution by convolution — starting from a single die and folding in one more at a time — so every figure here is a precise fraction, not a simulation.

Why 7 is the most common roll on 2d6

There are 36 equally likely outcomes, and six of them total 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1). Only one totals 2, and only one totals 12. The middle of the range simply has more ways to be reached, which is why craps is built around 7 and why the sum of many dice starts looking like a bell curve — the central limit theorem in its most tangible form.

E[sum] = n × (s + 1) / 2 Var[sum] = n × (s² − 1) / 12

Sums are not uniform, even though each die is

A single die is flat: every face equally likely. Add a second and the flatness disappears immediately. This catches people out in game design constantly — 3d6 for ability scores clusters hard around 10–11, while a d20 gives every result the same weight.

“At least” beats “exactly”

For most game questions the useful figure is a cumulative one — the chance of rolling at least a certain total, not exactly it. Switch the comparison above to see the tail probability directly.