4d10 Dice Probability — Full Distribution and Odds
Four ten-sided dice are the point where the shape looks smooth enough to use a normal approximation without much error. The sum runs from 4 to 40, the average is 22, and the most likely total is 22. 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 |
|---|---|---|---|---|---|
| 4 | 1 | 0.010% | 0.010% | 100.0% | 1 in 10000 |
| 5 | 4 | 0.040% | 0.050% | 100.0% | 1 in 2500 |
| 6 | 10 | 0.100% | 0.150% | 100.0% | 1 in 1000 |
| 7 | 20 | 0.200% | 0.350% | 99.9% | 1 in 500 |
| 8 | 35 | 0.350% | 0.700% | 99.7% | 1 in 286 |
| 9 | 56 | 0.560% | 1.26% | 99.3% | 1 in 179 |
| 10 | 84 | 0.840% | 2.10% | 98.7% | 1 in 119 |
| 11 | 120 | 1.20% | 3.30% | 97.9% | 1 in 83.3 |
| 12 | 165 | 1.65% | 4.95% | 96.7% | 1 in 60.6 |
| 13 | 220 | 2.20% | 7.15% | 95.0% | 1 in 45.5 |
| 14 | 282 | 2.82% | 9.97% | 92.8% | 1 in 35.5 |
| 15 | 348 | 3.48% | 13.5% | 90.0% | 1 in 28.7 |
| 16 | 415 | 4.15% | 17.6% | 86.6% | 1 in 24.1 |
| 17 | 480 | 4.80% | 22.4% | 82.4% | 1 in 20.8 |
| 18 | 540 | 5.40% | 27.8% | 77.6% | 1 in 18.5 |
| 19 | 592 | 5.92% | 33.7% | 72.2% | 1 in 16.9 |
| 20 | 633 | 6.33% | 40.1% | 66.3% | 1 in 15.8 |
| 21 | 660 | 6.60% | 46.7% | 60.0% | 1 in 15.2 |
| 22 | 670 | 6.70% | 53.3% | 53.3% | 1 in 14.9 |
| 23 | 660 | 6.60% | 60.0% | 46.7% | 1 in 15.2 |
| 24 | 633 | 6.33% | 66.3% | 40.1% | 1 in 15.8 |
| 25 | 592 | 5.92% | 72.2% | 33.7% | 1 in 16.9 |
| 26 | 540 | 5.40% | 77.6% | 27.8% | 1 in 18.5 |
| 27 | 480 | 4.80% | 82.4% | 22.4% | 1 in 20.8 |
| 28 | 415 | 4.15% | 86.6% | 17.6% | 1 in 24.1 |
| 29 | 348 | 3.48% | 90.0% | 13.5% | 1 in 28.7 |
| 30 | 282 | 2.82% | 92.8% | 9.97% | 1 in 35.5 |
| 31 | 220 | 2.20% | 95.0% | 7.15% | 1 in 45.5 |
| 32 | 165 | 1.65% | 96.7% | 4.95% | 1 in 60.6 |
| 33 | 120 | 1.20% | 97.9% | 3.30% | 1 in 83.3 |
| 34 | 84 | 0.840% | 98.7% | 2.10% | 1 in 119 |
| 35 | 56 | 0.560% | 99.3% | 1.26% | 1 in 179 |
| 36 | 35 | 0.350% | 99.7% | 0.700% | 1 in 286 |
| 37 | 20 | 0.200% | 99.9% | 0.350% | 1 in 500 |
| 38 | 10 | 0.100% | 100.0% | 0.150% | 1 in 1000 |
| 39 | 4 | 0.040% | 100.0% | 0.050% | 1 in 2500 |
| 40 | 1 | 0.010% | 100.0% | 0.010% | 1 in 10000 |
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 |
|---|---|---|
| ≥ 16 | 86.6% | 1 in 1.16 |
| ≥ 19 | 72.2% | 1 in 1.39 |
| ≥ 23 | 46.7% | 1 in 2.14 |
| ≥ 27 | 22.4% | 1 in 4.46 |
| ≥ 30 | 9.97% | 1 in 10 |
Reading the shape
Each die is uniform, but the sum is not. 4d10 has 10,000 ordered outcomes spread over 37 totals, and the counts rise toward the middle because a middle total can be assembled from more orderings. The curve is symmetric around 22 — for every way to roll 21 there is a mirrored way to roll 23 — which is why the standard deviation of 5.74 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 4d10?
22. The formula is dice × (faces + 1) / 2 — 4 × (10 + 1) / 2 — which works for any fair die. The standard deviation is 5.74, so most rolls land between 16 and 28.
What is the most likely total on 4d10?
Exactly 22, at 6.70% — 670 of the 10,000 equally likely outcomes. The extremes (4 and 40) take 1 way each, or 0.010%.
How many possible outcomes does 4d10 have?
10,000 ordered outcomes — 104 — collapsing into 37 possible sums from 4 to 40. 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 4d10 has a single peak at 22 and why adding dice makes the distribution tighter rather than wider.