Rolling Dice Fairly
This roller draws from crypto.getRandomValues, your browser's cryptographically secure random source, and uses rejection sampling so that every face is exactly equally likely. Naively taking a random number modulo six makes low values very slightly more common; discarding out-of-range draws removes that bias entirely.
Nothing is transmitted. The rolls happen on your device and are gone when you close the page.
Standard Dice
| Die | Faces | Shape | Average roll |
|---|---|---|---|
| D4 | 4 | Tetrahedron | 2.5 |
| D6 | 6 | Cube | 3.5 |
| D8 | 8 | Octahedron | 4.5 |
| D10 | 10 | Pentagonal trapezohedron | 5.5 |
| D12 | 12 | Dodecahedron | 6.5 |
| D20 | 20 | Icosahedron | 10.5 |
| D100 | 100 | Two D10s, or a zocchihedron | 50.5 |
Five of these are the Platonic solids — the only convex shapes with identical faces, which is what makes them provably fair. The D10 is not a Platonic solid, which is why it has a slightly odd kite-shaped face.
The average of a single die is always (sides + 1) ÷ 2, and for n dice it is n times that.
Why 2d6 Is Not Flat
One die gives a uniform distribution: every face equally likely. Two dice do not. There is only one way to roll a 2 and six ways to roll a 7, so 7 comes up six times as often.
| Total | Ways | Probability |
|---|---|---|
| 2 | 1 | 2.78% |
| 3 | 2 | 5.56% |
| 4 | 3 | 8.33% |
| 5 | 4 | 11.11% |
| 6 | 5 | 13.89% |
| 7 | 6 | 16.67% |
| 8 | 5 | 13.89% |
| 9 | 4 | 11.11% |
| 10 | 3 | 8.33% |
| 11 | 2 | 5.56% |
| 12 | 1 | 2.78% |
Add more dice and the distribution grows more sharply peaked, converging on a bell curve. This is the central limit theorem in its most tangible form, and it is why 3d6 produces far more moderate results than 1d18 despite a similar range.
Advantage and Disadvantage
Popularised by fifth-edition Dungeons & Dragons: roll twice and keep the higher (advantage) or lower (disadvantage). On a D20 the effect is substantial and non-uniform — it helps most in the middle of the range.
| Target | Normal | Advantage | Disadvantage |
|---|---|---|---|
| Roll 5+ | 80% | 96% | 64% |
| Roll 10+ | 55% | 80% | 30% |
| Roll 11+ | 50% | 75% | 25% |
| Roll 15+ | 30% | 51% | 9% |
| Roll 20 | 5% | 9.75% | 0.25% |
Advantage is worth roughly +5 on the roll when the target is near the middle, and much less at the extremes. The average D20 roll rises from 10.5 to 13.825 with advantage, and falls to 7.175 with disadvantage.
Dice Notation
The standard shorthand is NdS+M: N dice of S sides, plus a modifier M.
| Notation | Meaning | Range | Average |
|---|---|---|---|
| 1d6 | One six-sided die | 1–6 | 3.5 |
| 2d6 | Two six-sided dice | 2–12 | 7 |
| 1d20+5 | One D20 plus 5 | 6–25 | 15.5 |
| 3d6 | Three D6, the classic stat roll | 3–18 | 10.5 |
| 4d6 drop lowest | Roll four, discard the worst | 3–18 | 12.24 |
| 8d6 | Fireball damage | 8–48 | 28 |
Are Physical Dice Fair?
Not perfectly. Cheap dice with drilled and painted pips have slightly unequal mass distribution, since the 6 face has six cavities and the 1 face has one. Casino dice avoid this by using flush inlays of equal-density material, and they are made to tolerances of about 0.0005 inches with sharp edges rather than rounded ones.
The bias in ordinary dice is small — well under a percent per face — and undetectable in normal play, but it is real. A digital roller has no such problem.
The Gambler's Fallacy
Dice have no memory. After four sixes in a row, the chance of a fifth is exactly 1 in 6, the same as always. Streaks are what randomness actually produces: in 100 rolls of a D6, a run of three identical results is more likely than not.
Frequently Asked Questions
Are these rolls really random?
They come from your browser's cryptographic generator, seeded by operating-system entropy, with modulo bias removed. For any practical purpose they are indistinguishable from fair physical dice — and rather fairer than most cheap ones.
What does 4d6 drop lowest give?
An average of 12.24 rather than 10.5, which is why it is the standard method for generating heroic characters. Choose "Drop the lowest die" and roll four D6.
How much is advantage worth?
About +5 on a D20 near the middle of the range, and progressively less toward either extreme. It roughly doubles the chance of a natural 20.
Why is 7 the most common roll on two dice?
Because there are six ways to make it — 1+6, 2+5, 3+4 and their reverses — against a single way to make 2 or 12. Totals in the middle simply have more combinations.