Dice Roller & Probability Calculator
Roll any dice expression such as 4d6kh3+2, see each die, and get the exact probability of every total. It supports keep and drop rules, modifiers, exploding dice and a 10,000-roll simulation.
Written by Suhaib Hassan. Figures are estimates you can edit. Our methodology.
Rolls use the browser's secure random generator with rejection sampling. The odds are counted exactly for plain sums and for keep or drop rules on small pools.
Tabletop players want a quick roll and also want to know how likely a result was. This gives both together.
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Common dice compared
Exact figures for the expressions used most in tabletop games. The last column is the chance of rolling the highest possible total.
| Roll | Min | Max | Mean | Std dev | Variance | P(max) |
|---|---|---|---|---|---|---|
| 1d4 | 1 | 4 | 2.500 | 1.118 | 1.250 | 25.000% |
| 1d6 | 1 | 6 | 3.500 | 1.708 | 2.917 | 16.667% |
| 1d8 | 1 | 8 | 4.500 | 2.291 | 5.250 | 12.500% |
| 1d10 | 1 | 10 | 5.500 | 2.872 | 8.250 | 10.000% |
| 1d12 | 1 | 12 | 6.500 | 3.452 | 11.917 | 8.333% |
| 1d20 | 1 | 20 | 10.500 | 5.766 | 33.250 | 5.000% |
| 2d6 | 2 | 12 | 7.000 | 2.415 | 5.833 | 2.778% |
| 3d6 | 3 | 18 | 10.500 | 2.958 | 8.750 | 0.463% |
| 4d6 keep highest 3 | 3 | 18 | 12.245 | 2.847 | 8.105 | 1.620% |
| 2d20 keep highest 1 | 1 | 20 | 13.825 | 4.711 | 22.194 | 9.750% |
Formulas
How the numbers are worked out.
Mean and variance
For one die with s sides, then added up for n dice.
mean = n × (s + 1) ÷ 2
variance = n × (s² − 1) ÷ 12Exact probability of a total
Counted by combining the dice one at a time. With keep or drop rules, every combination is counted when there are few, otherwise the chart is simulated.
P(total = k) = ways to make k ÷ snFair random rolls
Rolls use the browser's secure random generator. Values that would bias the result are discarded and redrawn.
roll = (random mod s) + 1, redraw if in the biased tailWorked examples
Illustrative figures, each checked by hand. The first is the default setup.
Ability score
4d6, drop the lowest (4d6kh3), plus 2 (the default).
range 3 to 18 before the modifier, 5 to 20 after
Advantage
Roll two d20 and keep the higher.
P(20) = 1 − (19/20)² = 9.75%
Fireball
8d6.
std dev = √(8 × 35 ÷ 12) = 4.83
How to use the dice roller
- Type an expression such as 2d6+3, or tap dice and presets to build one.
- Roll the dice. Dropped dice are shown struck through.
- Read the expected value, chance of this result, and the probability chart.
- Run 10,000 rolls to compare the simulation with the exact odds.
Rolls are random. Past results do not change future odds.
By Suhaib Hassan · Reviewed 29 September 2026 · How CalculatePilot verifies formulas →
Sources
- Feller, W., An Introduction to Probability Theory and Its Applications, vol. 1, ch. 1 and 6.
- MDN Web Docs, Crypto.getRandomValues().
- NIST, SP 800-90A, Recommendation for Random Number Generation.
Frequently asked questions
What does 4d6kh3 mean?
Roll four six-sided dice and keep the highest three. It is a common way to generate character ability scores, giving an average of about 12.24.
What is advantage and disadvantage?
Advantage is rolling two d20 and keeping the higher (2d20kh1). Disadvantage keeps the lower (2d20kl1).
Are the rolls truly random?
They come from the browser's cryptographic random source, which is suitable for games. Values that would cause modulo bias are discarded.
What are exploding dice?
When a die shows its highest number you roll it again and add the result, repeating while it keeps showing the maximum.
Why do sums cluster in the middle?
There are more ways to make middle totals. With 2d6, seven can be made six ways while two can be made only one way.
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