How to use the calculator
Choose dice with the picker or type any roll in the notation box — 3d6, 2d20kh1, 1d20+1d4+5.
The Probability panel immediately shows a chart of every possible total, the range, average, most likely result and
standard deviation. Enter a target in Chance of rolling at least to get the odds of meeting it, and open the full table for the
exact, at-least and at-most chance of every total. Press Roll at any time to see where a real throw lands on the chart.
How the odds are calculated
Nothing here is estimated by simulation when it can be counted exactly:
- Sums of dice are worked out by convolution: the distribution for one die is combined with the next, one die at a time, so 3d6 is built from 2d6 and one more d6. This is exact for any number of dice and any mix of sizes.
- Keep and drop rolls (advantage, 4d6 drop lowest) are counted by visiting every distinct sorted outcome once and weighting it by the number of orders it can appear in — 126 cases for 4d6 instead of 1,296. Only for very large keep rolls, where even that would be too many, does the calculator fall back to 200,000 simulated rolls, and it says so.
- Modifiers shift the whole distribution; subtracted dice mirror it.
Average and spread of common rolls
For a sum of n dice with s sides, the average is n(s + 1)/2 and the standard deviation is √(n(s² − 1)/12). Roughly two thirds of rolls land within one standard deviation of the average.
| Roll | Range | Average | Std. deviation |
|---|---|---|---|
| 1d4 | 1–4 | 2.5 | 1.12 |
| 1d6 | 1–6 | 3.5 | 1.71 |
| 2d6 | 2–12 | 7 | 2.42 |
| 3d6 | 3–18 | 10.5 | 2.96 |
| 1d8 | 1–8 | 4.5 | 2.29 |
| 1d10 | 1–10 | 5.5 | 2.87 |
| 1d12 | 1–12 | 6.5 | 3.45 |
| 1d20 | 1–20 | 10.5 | 5.77 |
| 8d6 | 8–48 | 28 | 4.83 |
| 2d20kh1 (advantage) | 1–20 | 13.83 | 4.71 |
| 2d20kl1 (disadvantage) | 1–20 | 7.18 | 4.71 |
| 4d6dl1 (ability score) | 3–18 | 12.24 | 2.85 |
Why more dice make a bell curve
A single die is flat: every face is equally likely. Add a second and the middle totals can be made in more ways than the extremes, so the shape becomes a triangle peaking at 7. With three or more dice it rounds into a bell curve, and the more dice you add the closer it gets to the normal distribution — the central limit theorem at work. That is why games use 3d6 when they want average results to be common and extreme ones rare, and a single d20 when they want every result to be equally possible.
Worked examples
- 10 or more on 2d6: 10, 11 and 12 can be made 3 + 2 + 1 = 6 ways out of 36, so 6/36 = 16.67%.
- At least one 6 in four rolls of a die: 1 − (5/6)4 = 51.77%. This is the bet the Chevalier de Méré won with, and his losing variant — a double six in 24 throws of two dice (49.14%) — led Blaise Pascal and Pierre de Fermat to found probability theory in their letters of 1654.
- At least one natural 20 in three attack rolls: 1 − 0.953 = 14.26%.
- Hitting DC 15 with +4: you need 11 or more on the die — 10 faces of 20, so 50%; with advantage 1 − 0.5² = 75%.