Dice Roller
Roll any number of dice with any number of sides, using cryptographic randomness — with the probability facts that surprise most players.
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Four dice where “which is best” has no answer
Number four dice like this, roll two against each other, and the higher number wins:
| Die | Faces | Average |
|---|---|---|
| A | 4 4 4 4 0 0 | 2.67 |
| B | 3 3 3 3 3 3 | 3.00 |
| C | 6 6 2 2 2 2 | 3.33 |
| D | 5 5 5 1 1 1 | 3.00 |
Enumerating all 36 face pairs for each matchup — no simulation, no sampling:
| Matchup | Wins | Share |
|---|---|---|
| A beats B | 24 of 36 | 66.7% |
| B beats C | 24 of 36 | 66.7% |
| C beats D | 24 of 36 | 66.7% |
| D beats A | 24 of 36 | 66.7% |
Every arrow points the same way round the circle, every one at exactly two thirds, and not a single tie anywhere. There is no best die. Whichever you pick, the person choosing after you has one that beats it two times in three — which is why letting your opponent choose first looks generous and is not.
The instinct that resists this is assuming “wins more often” has to be an ordering, like taller or heavier. It does not. And averaging the faces to rank them does not rescue it: the averages are 2.67, 3.00, 3.33, 3.00, so A has the lowest of all four and still beats B, while C has the highest and still loses to it. Across the four matchups the higher average wins exactly twice — no better than guessing.
And rolling two of each destroys it
Roll two copies of each die and compare the totals, and the cycle does not turn around — it disappears. Two of the four original arrows fail, and what is left is a perfectly ordinary ranking:
| Die | Beats, with two of each |
|---|---|
| A | nothing |
| B | A |
| C | A, B, D |
| D | A |
C beats all three, A beats none, and B and D finish exactly level. So the non-transitivity is not a robust property of these numbers — it is a property of comparing them one roll at a time, and it does not survive being asked twice.
How to use
- Choose the die type and how many to roll.
- Add a modifier if your game needs one.
- Read the individual results and the total.
- Check the distribution if the results feel wrong.
Frequently asked questions
Is this actually random?
It uses the browser's cryptographic random source, which is unpredictable in practice and free of the bias that naive implementations introduce. It is not a physical die, but the distinction has no practical consequence for any game.
Why does rolling two dice not give a flat distribution?
Because there are more ways to make a middle total than an extreme one. Two six-sided dice make 7 in six different ways and 12 in only one, so 7 comes up six times as often. This is why board games using 2d6 feel so different from those using a single die.
What is modulo bias?
A subtle flaw where taking a random number modulo the number of sides makes low faces slightly more likely, because the range rarely divides evenly. A correct implementation rejects and retries the values that would skew it, which is what this one does.
Are physical dice fair?
Less often than people assume. Cheap moulded dice have hollowed pips that shift the centre of mass, and rounded corners vary. Casino dice are made to tight tolerances with flush pips precisely because ordinary dice are measurably biased.
What does 'advantage' mean in role-playing games?
Rolling two dice and keeping the higher, which shifts the average up substantially — for a twenty-sided die it is worth roughly a plus-5 bonus in the middle of the range, though the effect is largest where you need a middling result and smallest at the extremes.
Can I trust a digital roller for a real game?
For friendly play, yes, and it removes arguments about a die that fell off the table. For anything competitive, the group should agree in advance — the point of physical dice is that everyone can see them, which no screen provides.
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