Dice Roller

ways(total) = the coefficient of x^total in (x + x² + … + x^sides)^diceCount → chance = ways(total) / sides^diceCount

diceCount

How many dice are rolled, up to 20. One die gives a flat distribution where every face is equally likely; two or more pile the middle totals up, which is the whole reason the chance of a total is worth reporting

sides

How many faces each die has, from 2 to 100. The familiar d6 lives here, but so do the twenty-sided die and the hundred-sided one

ways(total)

How many of the possible outcomes add up to that total. The count of ways is the honest unit for dice — a total of 7 has six ways out of 36 with two six-sided dice, and that ratio is the probability

seed

The starting point of the roll, any whole number from 0 to 4294967295. The same seed with the same pool gives the same faces, which makes a roll something you can re-run and check

total

The sum of the faces. It runs from diceCount (all ones) to diceCount × sides (all top faces), so the number of totals a pool can make is diceCount × sides − diceCount + 1

Reach for it when the total matters and the individual faces are a detail: working out whether a 3d6 roll is likely to beat a target number, checking the odds of a 2d6 sum before a game, or producing a seeded roll so that a session can be replayed exactly. The seeded roll is the part that makes it more than a novelty — programs generating levels, groups replaying a scenario and people testing a rule all need a roll that can be reproduced, and a seed is what turns one roll into a roll anyone can reproduce. It is not a fair-dice simulator in the sense of physical randomness: the numbers come from a small deterministic sequence, so anything where a player must not be able to predict the faces is out of scope.

1. Two six-sided dice, seed 1: 4 and 1, for a total of 5

   1. The pool is two six-sided dice, so there are 6 × 6 = 36 equally likely pairs of faces
   2. Seed 1 fixes the pair as 4 and 1, and the total is 4 + 1 = 5
   3. A total of 5 comes up as 1+4, 2+3, 3+2 and 4+1 — four ways out of 36
   4. 4 / 36 = 0.1111…, so the chance is 11.11%

   The 11.11% is the point of putting the roll next to the odds: a total of 5 looks unremarkable, and it is the joint third-most likely total, but it still only happens about one roll in nine. Totals of 2 and 12 are the rarest at 2.78% each, because there is exactly one way to make each of them, while 7 has six ways and 16.67%. With a single die none of this structure exists — every face is 1/sides — which is why the pool size is the first thing to look at when the chance surprises you.

2. Three six-sided dice, seed 5: 5, 5 and 2, for a total of 12

   1. Three six-sided dice give 6 × 6 × 6 = 216 equally likely outcomes
   2. Seed 5 gives the faces 5, 5 and 2, which add to 12
   3. The pool can make 3 × 6 − 3 + 1 = 16 different totals, from 3 to 18
   4. A total of 12 has 25 ways out of 216, so the chance is 11.57%

   Three dice make the middle of the range much more crowded than two: 10 and 11 are the most likely totals at 12.5% each, and 12 is a close third at 11.57%. The same total of 12 is a very different proposition with two dice, where it is the rarest result of all at 2.78% — so a chance quoted for a total is meaningless unless the pool is quoted with it. The 16 possible totals also show why the count is not simply the number of faces: with three dice the totals run from 3 to 18 and there are 16 of them, not 18 and not 6.

3. One twenty-sided die, seed 7

   1. A single die has no way to combine faces, so every face is equally likely
   2. Seed 7 gives the face 1
   3. One face out of 20 is 1 / 20 = 5%

   This is the case where the distribution is flat: nothing piles up in the middle because there is only one die, so a 1 and a 20 are exactly as likely as anything else. It is the clearest way to see what the extra dice are doing — with 2d6 the extreme totals are rare, with 1d20 nothing is rare, and the mechanism is entirely the number of ways each total can be made. The random number generator next door makes the same point from the other direction: a single uniform draw is not the same thing as a sum of draws.

4. Seed at its upper limit

   1. The seed can go up to 4294967295, which is 2³² − 1 — the largest value the generator's state can hold
   2. The same two dice as the first example, but a different seed, so a different hand
   3. 6 and 2 add to 8, and a total of 8 has five ways out of 36
   4. 5 / 36 = 13.89%

   A seed of 4294967295 is worth trying because it is the one seed value where a careless implementation of the generator's state breaks in a way that is hard to see: the state is kept as a 32-bit unsigned integer, and the largest value is exactly where an off-by-one in that step stops being invisible. The visible result here is an ordinary hand of 6 and 2 — the panel is not supposed to look different — which is exactly why the boundary is a test and not a user-facing feature.

The faces are not unpredictable. They come from a small seeded sequence of the kind ordinary software uses, so anyone who knows the seed can reproduce the hand, and anyone who sees a few hands can work out the seed. That rules it out for anything where a player must not be able to predict or reconstruct the roll — a draw with money on it, a key, a token, a one-time code — and for those a cryptographic random source is the only correct answer. Seeded rolls are for replaying, generating and testing, where being reproducible is the point. Two further limits are worth stating plainly. The reference table below the calculator describes two six-sided dice and does not follow the dice you type in: the table is a fixed reference, and the chance on the result panel is the one that matches your pool. And the chance of a total assumes fair dice — a physical die with a bias will not match these numbers, whatever the pool.

Why does the table under the calculator not match the dice I entered?

Because the reference table is fixed to two six-sided dice. It is computed from a constant rather than from the inputs, so it stays put while you change the pool — the chance shown on the result panel is the one that follows your dice, and it is the number to read when the two differ. The table is there because 2d6 is the distribution people actually mean when they ask about dice odds: 7 in the middle, 2 and 12 at the edges. If you need the odds for a different pool, the panel gives the chance of the total you rolled, and you can roll the same seed again to check any other total you care about.

What does the seed do, and why is the same hand returned every time?

The seed is the state the roll starts from, so the same seed with the same pool always gives the same faces. That is the feature, not a shortcut: a seeded roll is how a session gets replayed exactly, how a generated level stays the same between runs, and how a rule can be tested against a fixed hand. Change the seed and you get a different hand from the same pool. If you want each roll to differ, change the seed between rolls — nothing in the panel quietly advances it for you.

Are these rolls suitable for a game where nobody may predict the result?

No, and the distinction matters. The faces come from a small deterministic sequence, so from the seed the whole hand can be reproduced, and from a few hands the seed can be recovered. Anything where a participant must not be able to predict or reconstruct the roll needs a cryptographic random source instead. Tabletop replay, level generation, rule testing and reproducible sampling are all fine — those are exactly the cases where being able to reproduce the roll is the reason to use it.

Why is a total of 7 the most likely with two dice, but not with three?

Because what piles up in the middle is the number of ways a total can be made, and that depends on the pool. With two six-sided dice there are 36 pairs and 7 has six of them (1+6 through 6+1), more than any other total. With three dice the most likely totals are 10 and 11 at 12.5% each, and 7 drops well down the list. The same total therefore means different things for different pools, which is why the panel always reports the total together with the chance for the pool you actually set.

How many totals can a pool make?

From the smallest possible total, which is all ones, to the largest, which is every die showing its top face: with a count of dice and sides per die that gives count × sides − count + 1 different totals. For 2d6 that is 11 totals running from 2 to 12; for 3d6 it is 16 running from 3 to 18; for 20d100 it is 1981. The count is not the number of faces, and the extremes of the range are always the rarest totals, because each of them can be made in exactly one way.

Is the chance on the panel for the total, or for the exact faces?

For the total. Two dice showing 4 and 1 and two dice showing 1 and 4 are the same total of 5, and the chance reported is the chance of that total, counting every combination that adds up to it — four ways out of 36 for a 5. The exact ordered hand is a different and much smaller probability (1/36 for any specific pair), and it is not what the panel reports, because the total is the number that decides whether a roll beats a target.

• Dice — from Wolfram MathWorld (the generating-function form of the sum of dice, and the coefficient that counts the ways each total can be made) — Wolfram MathWorld
• Uniform Sum Distribution — from Wolfram MathWorld (the distribution of a sum of uniform draws, which is what the total of several dice is) — Wolfram MathWorld
• 1.3.6.1. What is a Probability Distribution — e-Handbook of Statistical Methods (the frequency distribution of a variable, and how a probability is read off the outcomes it counts) — National Institute of Standards and Technology (NIST)