Dice roller
Any number of dice with any number of sides, written as fields or as notation, with the exact chance of the total.
Dice notation
The notation field reads the standard tabletop shorthand. NdS means N dice with S sides each, an optional keep or drop part chooses which dice count, and a number with + or − is added to the total. Spaces and capital letters are ignored, and the number of dice can be left out for a single die.
| Write | Meaning | Totals |
|---|---|---|
| d20 or 1d20 | One 20-sided die | 1–20 |
| 3d6+2 | Three 6-sided dice, add 2 | 5–20 |
| 1d8−1 | One 8-sided die, take away 1 | 0–7 |
| d% | Percentile die, the same as 1d100 | 1–100 |
| 2d20k1 (or 2d20kh1) | Two d20, keep the highest 1: advantage | 1–20 |
| 2d20kl1 | Two d20, keep the lowest 1: disadvantage | 1–20 |
| 4d6d1 (or 4d6dl1, 4d6k3) | Four d6, drop the lowest 1 | 3–18 |
| 5d10dh2 | Five d10, drop the highest 2 (keep the lowest 3) | 3–30 |
So k means keep the highest, kl keep the lowest, d drop the lowest and dh drop the highest. The fields and the notation stay in step: changing the number of dice, the sides or the modifier rewrites the notation, and rolling from the notation fills in the fields. Dropped dice stay on screen, greyed and struck through, so you can see which ones counted.
How the odds are worked out
Every combination of faces is equally likely, so the chance of a total is the number of combinations that make it divided by the number of possible rolls, SN. Counting the combinations one by one is impossible for large rolls (100d100 has 10200 of them), so the roller builds the distribution one die at a time. Adding a die spreads each existing total evenly over the next S totals, a step called convolution:
one more diePnew(t) = [P(t − 1) + P(t − 2) + … + P(t − S)] ÷ S2d6, total 7: 6 ÷ 36
at least tP(t) + P(t + 1) + … + P(highest)2d6, 7 or more: 21 ÷ 36
With a running sum this takes a few milliseconds even for 100 dice with 1,000 sides, and the result is exact apart from the usual rounding of a computer’s arithmetic, far below the two decimal places shown. When the number of possible rolls is a billion or fewer, the chance is also given as a count, such as 27 in 216; above that it is shown as a percentage or as 1 in a large number.
Keep and drop rolls need a different method, because the total depends on which dice are highest. The roller works through the face values from the highest down, tracking how many dice have landed on that face or above and the total of the ones being kept, which again gives the exact chance of every total. It covers rolls up to sizes such as 20d100 keep 19 or 100d6 drop 1; for keep and drop rolls larger than that, the dice are still rolled but the odds are not shown.
Totals with two six-sided dice
Two dice give 36 equally likely pairs. The totals in the middle can be made more ways, which is why 7 is six times as likely as 2 or 12.
| Total | Ways | Chance | This total or more |
|---|---|---|---|
| 2 | 1 | 2.78% | 100% |
| 3 | 2 | 5.56% | 97.22% |
| 4 | 3 | 8.33% | 91.67% |
| 5 | 4 | 11.11% | 83.33% |
| 6 | 5 | 13.89% | 72.22% |
| 7 | 6 | 16.67% | 58.33% |
| 8 | 5 | 13.89% | 41.67% |
| 9 | 4 | 11.11% | 27.78% |
| 10 | 3 | 8.33% | 16.67% |
| 11 | 2 | 5.56% | 8.33% |
| 12 | 1 | 2.78% | 2.78% |
Three dice spread further: 10 and 11 are the most likely totals at 27 in 216 (12.5%) each, and 3 or 18 is a 1 in 216 chance (0.46%). Rolling four dice and dropping the lowest raises the average from 10.5 to 12.24 and makes 18 three and a half times as likely, 21 in 1,296 (1.62%).
Advantage and disadvantage with 2d20
Rolling two d20 and keeping the higher one (advantage) or the lower one (disadvantage) changes the odds a lot more than it seems. With advantage you miss a target only if both dice miss, so the chance of reaching t or more is 1 − ((t − 1) ÷ 20)2. With disadvantage both dice must reach it, so the chance is ((21 − t) ÷ 20)2.
| Roll at least | One d20 | Advantage (2d20k1) | Disadvantage (2d20kl1) |
|---|---|---|---|
| 2 | 95% | 99.75% | 90.25% |
| 5 | 80% | 96% | 64% |
| 8 | 65% | 87.75% | 42.25% |
| 11 | 50% | 75% | 25% |
| 13 | 40% | 64% | 16% |
| 15 | 30% | 51% | 9% |
| 18 | 15% | 27.75% | 2.25% |
| 20 | 5% | 9.75% | 0.25% |
The average roll is 10.5 on one d20, 13.825 with advantage and 7.175 with disadvantage. Near the middle of the range, advantage adds 25 percentage points; at the extremes it adds much less.
Common dice and what they are used for
| Die | Shape | Typical use |
|---|---|---|
| d4 | Tetrahedron, 4 triangular faces | Small damage rolls in tabletop role-playing games |
| d6 | Cube; opposite faces add up to 7 | Board games, backgammon, craps, war games, classroom probability |
| d8 | Octahedron, 8 triangular faces | Role-playing damage and hit dice |
| d10 | Pentagonal trapezohedron, 10 kite-shaped faces, usually numbered 0–9 | Dice pools in some role-playing systems; a pair makes percentile dice |
| d12 | Dodecahedron, 12 pentagonal faces | Role-playing damage; picking a month |
| d20 | Icosahedron, 20 triangular faces | The main test roll in many role-playing games |
| d100 | Usually two d10s, or one 100-sided ball | Percentages and random tables |
The d4, d6, d8, d12 and d20 are the five regular convex solids, the shapes on which every face is identical, which is why they make fair dice. The d10 is not one of them, but all its faces are still the same shape. Here any number of sides from 2 to 1,000 works, so a d3, d7 or d30 is as fair as a d6.
Percentile dice
A d100 roll is usually made with two ten-sided dice of different colours, one read as tens (00, 10 … 90) and one as units (0 … 9). Each of the 100 pairs is equally likely, and by the most common convention 00 and 0 together read as 100, so the result runs from 1 to 100 with a 1 in 100 chance each. Rolling 1d100 here gives the same distribution. A roll of 25 or lower, for example, has a 25% chance, which is why percentile dice are used to check percentage chances directly.
Can a real die be biased?
Yes, slightly, and it has been measured. In 1894 the biologist W. F. R. Weldon recorded 26,306 throws of 12 dice, counting a 5 or 6 as a success. Fives and sixes came up 33.77% of the time instead of 33.33%, and Karl Pearson used the data in 1900 to introduce his chi-squared test, concluding that the dice were biased. The usual explanation is pip weight: on dice with hollowed-out pips the six face loses the most material, so it is the lightest and tends to end up on top.
In 2009 Zacariah Labby repeated the experiment with a dice-rolling machine and 12 cheap plastic dice, again 26,306 throws. Fives and sixes together were close to fair at 33.43%, but single faces were not: 1 came up 16.86% and 6 came up 16.88% of the time, against 16.67% expected. The cause was not pip weight but shape: the axis between the 1 and 6 faces was about 0.2% shorter than the other two, which makes those faces larger. Casino dice avoid the pip problem by filling the pips with material of the same density as the body.
A software die has no shape or weight, so none of this applies. Its fairness depends on the generator and on how random bits are turned into a face; this roller uses the browser’s cryptographic generator and rejection sampling, so each face of a d6 has exactly a 1 in 6 chance. The guide to fair shuffling and modulo bias explains the second step.
Questions
Is the dice roller fair?
Yes. Each die is a whole number from 1 to the number of sides, drawn from your browser’s cryptographic random generator by rejection sampling, so every face has exactly the same chance and no die affects another. Physical dice can’t promise that; the section on biased dice above shows by how much real dice have been measured to drift.
Why does 7 come up most often with two dice?
Because more combinations make 7 than any other total. Six of the 36 equally likely pairs add up to 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1), while 2 and 12 can each be made only one way. Each die on its own is still equally likely to show any face.
What does 4d6 drop lowest mean?
Roll four six-sided dice, ignore the lowest one and add up the other three. Write it as 4d6d1 or 4d6k3. It gives totals from 3 to 18 like 3d6, but higher on average, 12.24 instead of 10.5. If two dice tie for lowest, only one is dropped; the result is the same either way.
Can I roll different dice at once, such as a d8 and a d6?
Not in one roll; every die in a roll has the same number of sides, which is what makes the exact odds possible. Roll 1d8 and then 1d6 and add the totals; the log keeps both rolls.
Why does a previous roll not change the next one?
Dice have no memory. After three 1s in a row, the next roll of a d6 is still a 1 in 6 chance of each face. Streaks happen often in random sequences, which is why they feel meaningful.
Are my rolls saved?
Only on this page while it stays open. The log of rolls is kept in the page and disappears when you leave. The address bar holds the dice settings, not the results, so a link you share opens the same dice but rolls new numbers.