One rule for every die
The dice used in tabletop role-playing and board games come in several shapes, each named by its number of faces: d4, d6, d8, d10, d12 and d20. The “d” stands for die, so a d8 has eight faces. If a die is fair, each face has the same probability, which is 1 divided by the number of faces.
| Die | Faces | Chance of one given face | Average (faces numbered from 1) |
|---|---|---|---|
| d4 | 4 | 25% | 2.5 |
| d6 | 6 | about 16.7% | 3.5 |
| d8 | 8 | 12.5% | 4.5 |
| d10 | 10 | 10% | 5.5 |
| d12 | 12 | about 8.3% | 6.5 |
| d20 | 20 | 5% | 10.5 |
The average of a die numbered 1 to n is (n + 1) / 2. This is the midpoint of the range, and it is also the expected value of a single roll. No roll will give a value like 3.5 on a d6, but across many rolls the mean settles near it.
Shapes and fairness
Fair dice are built from solids in which every face is the same shape and size, so no face is naturally favoured by the geometry. The d4 is a tetrahedron, the d8 is an octahedron, the d12 is a dodecahedron and the d20 is an icosahedron. The d6 is a cube. The d10 is different: it is not one of the five regular solids, but a shape with ten kite-shaped faces that is designed so each face is equally likely.
Because the d4 has no top face when it lands, the result is read in a different way, such as from the number at the top corner or the bottom edge, depending on the die. Whichever method is used, the idea of equal probability stays the same.
The d10 and percentile dice
Many d10s are numbered 0 to 9 rather than 1 to 10. In that case the average is 4.5, not 5.5, unless the 0 is treated as a 10. Two d10s can be used together to produce a number from 1 to 100: one as tens and one as units, with 00 and 0 read as 100. Each of the 100 results has probability 1/100, which is 1 percent. This is why you will see percentile rolls described in some games.
A single d20 is flat
A single roll of any fair die has a uniform distribution: every result is equally likely, so a chart of the chances is a flat line. For a d20, each value from 1 to 20 has probability 5 percent. The chance of 1 is exactly the same as the chance of 10 or 20.
Some consequences follow directly from counting faces.
- The chance of rolling 15 or higher is 6 out of 20, which is 30 percent.
- The chance of rolling 10 or lower is 10 out of 20, which is 50 percent.
- Each step of 1 in a target number changes the chance by 5 percentage points.
That last point is a convenient feature of a d20: it makes it easy to see how much a +1 bonus is worth. A +1 to a d20 roll raises the chance of reaching a target by 5 percentage points, as long as the target stays between 2 and 20. Mixed rolls such as taking the higher of two dice are the subject of the article on advantage rolls.
What happens when you add dice
Flat spreads occur only for one die. When you add two or more dice, the totals are no longer equally likely. Two d6 give a triangular shape with 7 as the most common total, and three or more dice start to look like a bell curve. Two d4 behave the same way on a smaller scale: totals from 2 to 8, with 5 most likely at 4 ways out of 16. Different dice distributions can give a very different feel to a game, even when the averages are similar. A d20 gives wild swings; 3d6 gives more consistent results clustered near 10 or 11.
A designer choosing between a single d20 and, say, two d10s should remember the difference: a single roll is flat, while the sum of two dice favours middle values.
Common misunderstandings
A few mistakes come up often.
- Thinking a number is “due”. Each roll of a fair die is independent, and a d20 that has not shown a 20 for a long time is not more likely to do so. This is the gambler’s fallacy.
- Believing a particular die is “lucky”. Some players do have dice that feel lucky, but a fair die has no memory. Lucky rituals have no mathematical effect.
- Assuming all physical dice are perfect. Cheap or worn dice may have slight biases, which is why some players test dice by counting results.
Why it matters in games
Understanding these numbers helps with decisions in tabletop games, such as judging how risky an action is, and with understanding the design logic of game design randomness in general. It is also useful for spotting when someone is quoting a misleading percentage. Probabilities from dice describe chance, and a single roll can go any way.
Outside games, the same ideas appear in gambling, where payouts are set below the true odds. A house edge means that fair dice do not make a gamble fair. If gambling causes you difficulty, a local support service can help.