Dice probability explained
Dice are the easiest way to see probability working. One die is simple: every face is equally likely. Add a second die and something more interesting happens, because some totals can be made in more ways than others.
This guide gives the numbers you actually need at the table: the odds of each total on two dice, the average roll of every common die, what advantage really does to a d20, and a few classic board game odds.
One die: every face is equal
On a fair six-sided die each number has a 1 in 6 chance, about 16.67%. The same rule applies to any fair die: the chance of a given face is 1 divided by the number of sides. A d20 gives each number a 5% chance, which is why tabletop games use it. Every point of bonus is worth exactly 5%.
| Die | Chance of one face | Average roll |
|---|---|---|
| d4 | 25% | 2.5 |
| d6 | 16.67% | 3.5 |
| d8 | 12.5% | 4.5 |
| d10 | 10% | 5.5 |
| d12 | 8.33% | 6.5 |
| d20 | 5% | 10.5 |
| d100 | 1% | 50.5 |
The average is always half of the highest face plus a half. To estimate the total of several dice, multiply: three d6 average 10.5, and eight d6 average 28.
Two dice: why 7 comes up most
Two six-sided dice can land in 36 different ways, and every one of those 36 is equally likely. The totals are not. Only one combination makes 2 (a pair of ones), but six combinations make 7: 1 and 6, 2 and 5, 3 and 4, and each of those the other way round.
| Total | Ways to roll it | Chance |
|---|---|---|
| 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% |
This table explains a lot of board game design. In games where two dice decide which spaces pay out, the spaces numbered 6 and 8 are the most valuable after 7, and 2 and 12 almost never come up. You can test it yourself: set the Dice Roller to two d6, roll fifty times and tally the totals. The middle of the table fills up first.
Useful two-dice odds
- Rolling a double: 6 in 36, or 16.67%.
- Rolling a specific double, such as double six: 1 in 36, or 2.78%.
- At least one six on two dice: 11 in 36, or 30.56%.
- A total of 7 or 11: 8 in 36, or 22.22%.
- A total of 10 or more: 6 in 36, or 16.67%.
- Three doubles in a row: 1 in 216, or 0.46%.
Advantage and disadvantage on a d20
Many tabletop role-playing games use advantage: roll two d20s and keep the higher. Disadvantage is the reverse. The effect is bigger than most players assume, and it is strongest in the middle of the range.
| Normal roll | Advantage | Disadvantage | |
|---|---|---|---|
| Average result | 10.5 | 13.83 | 7.18 |
| Chance of 11 or higher | 50% | 75% | 25% |
| Chance of 15 or higher | 30% | 51% | 9% |
| Chance of a natural 20 | 5% | 9.75% | 0.25% |
| Chance of a natural 1 | 5% | 0.25% | 9.75% |
Advantage is worth roughly a +5 bonus when you need an 11, but far less when you need a 20 or only a 2. To roll with advantage here, set the Dice Roller to two d20s and read the higher die.
Classic dice questions
How likely is at least one six in four rolls? It is 1 minus the chance of no sixes at all, which is five-sixths multiplied by itself four times. The answer is 51.77%, slightly better than even. This is the bet that started the study of probability in the 1650s, when the gambler Antoine Gombaud asked Blaise Pascal why it paid off.
How likely is a five-of-a-kind in one roll of five dice? There are 7,776 possible rolls and six of them are five of a kind, so the chance is 1 in 1,296, about 0.08%. That is why games built around it give you re-rolls.
Are online dice as fair as real ones?
A well-made virtual die is at least as fair as a physical one. Real dice can be slightly unbalanced by drilled pips, rounded corners or air bubbles in the plastic, and a lazy roll that barely tumbles is not very random. A virtual die has none of those problems, provided it uses a good source of randomness.
Our dice roller uses the browser's cryptographically secure random number generator, and each die is drawn independently. For more on the difference between good and poor random sources, see how online wheel spinners work.
Frequently asked questions
What is the most likely total with two dice?
Seven. Six of the 36 possible combinations add up to 7, giving it a 16.67% chance, or 1 in 6.
What is the average roll of a d20?
The average is 10.5. With advantage it rises to about 13.83, and with disadvantage it falls to about 7.18.
What are the odds of rolling doubles?
One in six, or 16.67%. There are six doubles among the 36 possible results of two six-sided dice.
How much does advantage help on a d20?
It depends on the number you need. Needing an 11, advantage lifts your chance from 50% to 75%. Needing a 20, it only lifts it from 5% to 9.75%.