3d20 Dice Probability — Full Distribution and Odds
Rare in play but common in design work: 8,000 outcomes and a curve so regular it is nearly a normal distribution. The sum runs from 3 to 60, the average is 31.5, and the most likely total is 31. Below is every sum with its exact number of ways, its probability, the cumulative odds and the "at least" thresholds worth memorising.
Every possible sum
| Sum | Ways | Exactly | At most | At least | Odds of exactly |
|---|---|---|---|---|---|
| 3 | 1 | 0.013% | 0.013% | 100.0% | 1 in 8000 |
| 4 | 3 | 0.037% | 0.050% | 100.0% | 1 in 2667 |
| 6 | 10 | 0.125% | 0.250% | 99.9% | 1 in 800 |
| 7 | 15 | 0.188% | 0.438% | 99.8% | 1 in 533 |
| 9 | 28 | 0.350% | 1.05% | 99.3% | 1 in 286 |
| 10 | 36 | 0.450% | 1.50% | 99.0% | 1 in 222 |
| 12 | 55 | 0.688% | 2.75% | 97.9% | 1 in 145 |
| 13 | 66 | 0.825% | 3.57% | 97.3% | 1 in 121 |
| 15 | 91 | 1.14% | 5.69% | 95.5% | 1 in 87.9 |
| 16 | 105 | 1.31% | 7.00% | 94.3% | 1 in 76.2 |
| 18 | 136 | 1.70% | 10.2% | 91.5% | 1 in 58.8 |
| 19 | 153 | 1.91% | 12.1% | 89.8% | 1 in 52.3 |
| 21 | 190 | 2.38% | 16.6% | 85.8% | 1 in 42.1 |
| 22 | 210 | 2.63% | 19.3% | 83.4% | 1 in 38.1 |
| 23 | 228 | 2.85% | 22.1% | 80.8% | 1 in 35.1 |
| 25 | 258 | 3.23% | 28.4% | 74.9% | 1 in 31 |
| 26 | 270 | 3.38% | 31.8% | 71.6% | 1 in 29.6 |
| 28 | 288 | 3.60% | 38.9% | 64.8% | 1 in 27.8 |
| 29 | 294 | 3.67% | 42.5% | 61.2% | 1 in 27.2 |
| 31 | 300 | 3.75% | 50.0% | 53.8% | 1 in 26.7 |
| 32 | 300 | 3.75% | 53.8% | 50.0% | 1 in 26.7 |
| 34 | 294 | 3.67% | 61.2% | 42.5% | 1 in 27.2 |
| 35 | 288 | 3.60% | 64.8% | 38.9% | 1 in 27.8 |
| 37 | 270 | 3.38% | 71.6% | 31.8% | 1 in 29.6 |
| 38 | 258 | 3.23% | 74.9% | 28.4% | 1 in 31 |
| 40 | 228 | 2.85% | 80.8% | 22.1% | 1 in 35.1 |
| 41 | 210 | 2.63% | 83.4% | 19.3% | 1 in 38.1 |
| 42 | 190 | 2.38% | 85.8% | 16.6% | 1 in 42.1 |
| 44 | 153 | 1.91% | 89.8% | 12.1% | 1 in 52.3 |
| 45 | 136 | 1.70% | 91.5% | 10.2% | 1 in 58.8 |
| 47 | 105 | 1.31% | 94.3% | 7.00% | 1 in 76.2 |
| 48 | 91 | 1.14% | 95.5% | 5.69% | 1 in 87.9 |
| 50 | 66 | 0.825% | 97.3% | 3.57% | 1 in 121 |
| 51 | 55 | 0.688% | 97.9% | 2.75% | 1 in 145 |
| 53 | 36 | 0.450% | 99.0% | 1.50% | 1 in 222 |
| 54 | 28 | 0.350% | 99.3% | 1.05% | 1 in 286 |
| 56 | 15 | 0.188% | 99.8% | 0.438% | 1 in 533 |
| 57 | 10 | 0.125% | 99.9% | 0.250% | 1 in 800 |
| 59 | 3 | 0.037% | 100.0% | 0.050% | 1 in 2667 |
| 60 | 1 | 0.013% | 100.0% | 0.013% | 1 in 8000 |
Large pools are sampled to 40 rows so the table stays readable; the page numbers are still exact, not interpolated.
Rolling a target or higher
This is the form most checks actually use — "did I beat 12?" rather than "did I roll exactly 12".
| Target | Chance of meeting it | Odds |
|---|---|---|
| ≥ 19 | 89.8% | 1 in 1.11 |
| ≥ 25 | 74.9% | 1 in 1.34 |
| ≥ 32 | 50.0% | 1 in 2 |
| ≥ 40 | 22.1% | 1 in 4.52 |
| ≥ 46 | 8.50% | 1 in 11.8 |
Reading the shape
Each die is uniform, but the sum is not. 3d20 has 8,000 ordered outcomes spread over 58 totals, and the counts rise toward the middle because a middle total can be assembled from more orderings. The curve is symmetric around 31.5 — for every way to roll 30.5 there is a mirrored way to roll 32.5 — which is why the standard deviation of 9.99 describes the spread on both sides equally.
Compare that with a single die: 1d20 is a flat line at 5.00% per face. Adding dice does not widen the range proportionally — it makes extreme results rarer. That is the whole reason systems that want predictable outcomes roll pools, and systems that want drama roll one big die.
Frequently asked
What is the average roll of 3d20?
31.5. The formula is dice × (faces + 1) / 2 — 3 × (20 + 1) / 2 — which works for any fair die. The standard deviation is 9.99, so most rolls land between 21 and 42.
What is the most likely total on 3d20?
31 and 32, at 3.75% — 300 of the 8,000 equally likely outcomes. The extremes (3 and 60) take 1 way each, or 0.013%.
How many possible outcomes does 3d20 have?
8,000 ordered outcomes — 203 — collapsing into 58 possible sums from 3 to 60. The counts here are exact integers from a convolution, not a normal approximation, so the tails are right: a normal curve would misstate the odds of the extreme rolls.
Why is the middle so much more likely than the ends?
Because sums have more ways to happen than faces do. There is exactly one way to roll all ones, but a middle total can be assembled from many orderings, and each ordering is a distinct outcome. That is why 3d20 has a plateau at 31 and 32 and why adding dice makes the distribution tighter rather than wider.