Rolling dice, and what the odds of a total really are
How dice notation like 3d6 works, the exact odds for two and three dice and a d20, why the middle totals come up most, how advantage works, and how a seeded online dice roller differs from a real die.
Last reviewed · 1,400 words
In short
- Dice notation NdS means roll N dice with S sides and add them: 2d6 is two six-sided dice, and 1d20 + 3 is one twenty-sided die plus 3.
- With two six-sided dice there are 36 equally likely combinations, and 7 comes up 6 times in 36, which is 16.67 percent or 1 in 6; 2 and 12 each come up 1 time in 36.
- The more dice you add, the more the total bunches around the average: 3d6 averages 10.5, and 10 and 11 are each 12.5 percent likely while 3 and 18 are each 0.46 percent.
- Rolling two d20 and keeping the higher (advantage) lifts the chance of at least 15 from 30 percent to 51 percent.
- Each roll is independent: a die has no memory, so a long run without a six makes a six no more likely.
A die looks like the simplest source of chance there is: six faces, each equally likely. The interesting part begins when you roll more than one, or when a game asks for a particular total. Then the odds are no longer even, and working them out by hand gets tedious fast. This page explains dice notation, shows the exact odds for the most common rolls, and describes how the online roller works and how it differs from a real die.
Dice notation
Games shorthand dice as NdS: N dice, each with S sides. So 1d6 is one ordinary die, 2d6 is two of them added, and 3d6 is three. A number after a plus or minus sign is a modifier, added to the total once: 1d20 + 3 is one twenty-sided die with 3 added to the result.
The common dice are the d4 (a four-sided pyramid), the d6 (the cube), the d8, the d10, the d12, the d20, and the d100, which gives a percentage. Roleplaying games lean on the d20 for attacks and skill checks; board games use d6; a d100 stands in for a percentage chance.
The calculator above rolls up to ten dice with any of these numbers of sides, adds your modifier, and shows the total together with the individual dice. The range, the average roll, and the chance of the total you actually got are shown beside it.
One die: everything is equal
With a single fair die, every face has the same chance. On a d6 each number is 1 in 6, or 16.67 percent. On a d20 each is 1 in 20, or 5 percent. The average of a d6 is 3.5, because the faces 1 to 6 add up to 21 and 21 ÷ 6 = 3.5. In general the average of a die with S sides is (S + 1) ÷ 2, and of N such dice, N × (S + 1) ÷ 2. So 2d6 averages 7, 3d6 averages 10.5, and 1d20 averages 10.5.
Two dice: the middle wins
Add a second die and the situation changes. There are 6 × 6 = 36 ways two dice can land, all equally likely, but several give the same total. The number of ways to make each total is:
| Total | Ways | Chance | Total | Ways | Chance |
|---|---|---|---|---|---|
| 2 | 1 | 2.78% | 8 | 5 | 13.89% |
| 3 | 2 | 5.56% | 9 | 4 | 11.11% |
| 4 | 3 | 8.33% | 10 | 3 | 8.33% |
| 5 | 4 | 11.11% | 11 | 2 | 5.56% |
| 6 | 5 | 13.89% | 12 | 1 | 2.78% |
| 7 | 6 | 16.67% |
Seven is the most likely total because six different combinations make it: 1+6, 2+5, 3+4, and each reversed. A 2 can only be 1+1, and a 12 only 6+6. That is why seven is the number a board game hands to the most common event, and why a total of 2 or 12 is something to remember.
The chance of "at least" a number is just the sum of the rows. A total of 10 or more is 3 + 2 + 1 = 6 ways in 36, which is 16.67 percent, and 8 or more is 15 ways in 36, or 41.67 percent.
Three dice and the bell shape
With three dice there are 6³ = 216 combinations, and the totals bunch even more around the middle. The ways to make each total from 3 to 18 are 1, 3, 6, 10, 15, 21, 25, 27, 27, 25, 21, 15, 10, 6, 3 and 1. So 10 and 11 each come up 27 times in 216, which is 12.5 percent, while 3 and 18 each come up once, 0.46 percent.
This is a general rule. The more dice you add, the more the totals form a bell shape, thin at both ends and fat in the middle. It is why games that want a steady, predictable outcome roll several small dice, and games that want swings use one large die. A single d20 swings from 1 to 20 with equal chance, while 3d6 almost always lands between 6 and 15.
The spread is measured by the standard deviation. One d6 has a variance of 35 ÷ 12 = 2.92, and variances add for independent dice, so two dice have 5.83, a standard deviation of 2.42. The totals of 2d6 therefore stay mostly within about two and a half of the average of 7.
Advantage, disadvantage and other tricks
Roleplaying games vary the d20 in a way that is easy to misjudge. With advantage you roll two d20 and keep the higher. The chance of getting at least 15 on one die is 6 in 20, or 30 percent. With advantage you fail only if both dice fall below 15, which is 0.7 × 0.7 = 0.49, so the chance of at least 15 is 1 − 0.49 = 51 percent. With disadvantage, keeping the lower, you need both dice to reach 15: 0.3 × 0.3 = 9 percent.
Advantage on a roll of a natural 20 takes it from 5 percent to 1 − 0.95² = 9.75 percent.
Character generation in some games uses 4d6, drop the lowest. Rolling four dice and discarding the smallest gives an average of 12.24 per ability score, higher than the 10.5 average of plain 3d6, which is why heroes built this way are above average.
The gambler's fallacy
Dice have no memory. If a d6 has not shown a six in twenty rolls, the next roll is still a 1 in 6 chance, not better. It is true that the chance of no six at all in twenty rolls is small, (5/6)²⁰ = 2.6 percent, which is why a long drought feels remarkable. But once those twenty rolls have happened, they are in the past and the next roll starts fresh.
The reverse question is the one worth asking. The chance of at least one six in four rolls of a d6 is 1 − (5/6)⁴ = 51.77 percent, which is the famous bet of the seventeenth-century gambler Chevalier de Méré, who was right to take it. His other bet, at least one double six in 24 throws of two dice, is 1 − (35/36)²⁴ = 49.14 percent, slightly worse than even, and he lost money on it.
How this roller works
A physical die is random because of how it tumbles. This roller is computed, so it uses a pseudo-random generator (mulberry32), seeded from the roll number and the seed text. Move the roll number, or change the seed, and you get a different set of dice. The same roll number and the same seed always give the same dice.
That is a deliberate choice for two reasons. The page is prerendered on the server and then shown in your browser, and an unseeded random roll would give a different result in each place. And a roll you can repeat is useful: a game master can say "roll number 14", and anyone can check that it gave a 9. The cost is that a particular seed's sequence is fixed, but across seeds and roll numbers the results are spread evenly, as the tests on this site confirm: the average of thousands of d6 rolls sits at 3.5.
It is not a cryptographic generator, and it should not be used where money or security depends on the result. For a board game, a classroom exercise, a tabletop campaign or a quick decision, it is perfectly fair. For a draw of numbers, the random number generator offers ranges, duplicates and sorting.
The odds table under the result
The table under the roller lists the twelve most likely totals for your dice, with the chance of each and the same as "1 in N". For 2d6 it is the whole range. For ten dice with 100 sides it shows only the centre, because there are hundreds of possible totals and the extreme ones have chances too small to be useful.
Each figure is exact, calculated by combining the odds one die at a time, and it already includes any modifier you chose, so the table's totals match the totals you read.
What this calculator assumes
- Every face of every die is equally likely.
- Dice are independent of one another, and of earlier rolls.
- The number of dice is limited to ten, and the sides to 4, 6, 8, 10, 12, 20 or 100.
- The modifier is added to the total once and does not change the odds of the dice themselves.
- The roll is pseudo-random and repeatable from its roll number and seed, and is not suitable for security uses.