3d10 Dice Probability — Full Distribution and Odds
Storyteller-style pools and damage on big weapons, with a mean of 16.5 and a standard deviation near 5. The sum runs from 3 to 30, the average is 16.5, and the most likely total is 16. 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 | 0.100% | 0.100% | 100.0% | 1 in 1000 |
| 4 | 3 | 0.300% | 0.400% | 99.9% | 1 in 333 |
| 5 | 6 | 0.600% | 1.00% | 99.6% | 1 in 167 |
| 6 | 10 | 1.00% | 2.00% | 99.0% | 1 in 100 |
| 7 | 15 | 1.50% | 3.50% | 98.0% | 1 in 66.7 |
| 8 | 21 | 2.10% | 5.60% | 96.5% | 1 in 47.6 |
| 9 | 28 | 2.80% | 8.40% | 94.4% | 1 in 35.7 |
| 10 | 36 | 3.60% | 12.0% | 91.6% | 1 in 27.8 |
| 11 | 45 | 4.50% | 16.5% | 88.0% | 1 in 22.2 |
| 12 | 55 | 5.50% | 22.0% | 83.5% | 1 in 18.2 |
| 13 | 63 | 6.30% | 28.3% | 78.0% | 1 in 15.9 |
| 14 | 69 | 6.90% | 35.2% | 71.7% | 1 in 14.5 |
| 15 | 73 | 7.30% | 42.5% | 64.8% | 1 in 13.7 |
| 16 | 75 | 7.50% | 50.0% | 57.5% | 1 in 13.3 |
| 17 | 75 | 7.50% | 57.5% | 50.0% | 1 in 13.3 |
| 18 | 73 | 7.30% | 64.8% | 42.5% | 1 in 13.7 |
| 19 | 69 | 6.90% | 71.7% | 35.2% | 1 in 14.5 |
| 20 | 63 | 6.30% | 78.0% | 28.3% | 1 in 15.9 |
| 21 | 55 | 5.50% | 83.5% | 22.0% | 1 in 18.2 |
| 22 | 45 | 4.50% | 88.0% | 16.5% | 1 in 22.2 |
| 23 | 36 | 3.60% | 91.6% | 12.0% | 1 in 27.8 |
| 24 | 28 | 2.80% | 94.4% | 8.40% | 1 in 35.7 |
| 25 | 21 | 2.10% | 96.5% | 5.60% | 1 in 47.6 |
| 26 | 15 | 1.50% | 98.0% | 3.50% | 1 in 66.7 |
| 27 | 10 | 1.00% | 99.0% | 2.00% | 1 in 100 |
| 28 | 6 | 0.600% | 99.6% | 1.00% | 1 in 167 |
| 29 | 3 | 0.300% | 99.9% | 0.400% | 1 in 333 |
| 30 | 1 | 0.100% | 100.0% | 0.100% | 1 in 1000 |
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 |
|---|---|---|
| ≥ 11 | 88.0% | 1 in 1.14 |
| ≥ 14 | 71.7% | 1 in 1.39 |
| ≥ 17 | 50.0% | 1 in 2 |
| ≥ 21 | 22.0% | 1 in 4.55 |
| ≥ 24 | 8.40% | 1 in 11.9 |
Reading the shape
Each die is uniform, but the sum is not. 3d10 has 1,000 ordered outcomes spread over 28 totals, and the counts rise toward the middle because a middle total can be assembled from more orderings. The curve is symmetric around 16.5 — for every way to roll 15.5 there is a mirrored way to roll 17.5 — which is why the standard deviation of 4.97 describes the spread on both sides equally.
Compare that with a single die: 1d10 is a flat line at 10.0% 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 3d10?
16.5. The formula is dice × (faces + 1) / 2 — 3 × (10 + 1) / 2 — which works for any fair die. The standard deviation is 4.97, so most rolls land between 11 and 22.
What is the most likely total on 3d10?
16 and 17, at 7.50% — 75 of the 1,000 equally likely outcomes. The extremes (3 and 30) take 1 way each, or 0.100%.
How many possible outcomes does 3d10 have?
1,000 ordered outcomes — 103 — collapsing into 28 possible sums from 3 to 30. 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 3d10 has a plateau at 16 and 17 and why adding dice makes the distribution tighter rather than wider.