1d20 Dice Probability — Full Distribution and Odds
The d20 that decides almost every attack in modern D&D: 5% per face, and no bell curve to hide behind. The sum runs from 1 to 20, the average is 10.5, and the most likely total is 1. 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 |
|---|---|---|---|---|---|
| 1 | 1 | 5.00% | 5.00% | 100.0% | 1 in 20 |
| 2 | 1 | 5.00% | 10.0% | 95.0% | 1 in 20 |
| 3 | 1 | 5.00% | 15.0% | 90.0% | 1 in 20 |
| 4 | 1 | 5.00% | 20.0% | 85.0% | 1 in 20 |
| 5 | 1 | 5.00% | 25.0% | 80.0% | 1 in 20 |
| 6 | 1 | 5.00% | 30.0% | 75.0% | 1 in 20 |
| 7 | 1 | 5.00% | 35.0% | 70.0% | 1 in 20 |
| 8 | 1 | 5.00% | 40.0% | 65.0% | 1 in 20 |
| 9 | 1 | 5.00% | 45.0% | 60.0% | 1 in 20 |
| 10 | 1 | 5.00% | 50.0% | 55.0% | 1 in 20 |
| 11 | 1 | 5.00% | 55.0% | 50.0% | 1 in 20 |
| 12 | 1 | 5.00% | 60.0% | 45.0% | 1 in 20 |
| 13 | 1 | 5.00% | 65.0% | 40.0% | 1 in 20 |
| 14 | 1 | 5.00% | 70.0% | 35.0% | 1 in 20 |
| 15 | 1 | 5.00% | 75.0% | 30.0% | 1 in 20 |
| 16 | 1 | 5.00% | 80.0% | 25.0% | 1 in 20 |
| 17 | 1 | 5.00% | 85.0% | 20.0% | 1 in 20 |
| 18 | 1 | 5.00% | 90.0% | 15.0% | 1 in 20 |
| 19 | 1 | 5.00% | 95.0% | 10.0% | 1 in 20 |
| 20 | 1 | 5.00% | 100.0% | 5.00% | 1 in 20 |
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 |
|---|---|---|
| ≥ 3 | 90.0% | 1 in 1.11 |
| ≥ 6 | 75.0% | 1 in 1.33 |
| ≥ 11 | 50.0% | 1 in 2 |
| ≥ 16 | 25.0% | 1 in 4 |
| ≥ 19 | 10.0% | 1 in 10 |
Reading the shape
Each die is uniform, but the sum is not. 1d20 has 20 ordered outcomes spread over 20 totals, and the counts rise toward the middle because a middle total can be assembled from more orderings. The curve is symmetric around 10.5 — for every way to roll 9.5 there is a mirrored way to roll 11.5 — which is why the standard deviation of 5.77 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 1d20?
10.5. The formula is dice × (faces + 1) / 2 — 1 × (20 + 1) / 2 — which works for any fair die. The standard deviation is 5.77, so most rolls land between 4 and 17.
What is the most likely total on 1d20?
1 and 2, at 5.00% — 1 of the 20 equally likely outcomes. The extremes (1 and 20) take 1 way each, or 5.00%.
How many possible outcomes does 1d20 have?
20 ordered outcomes — 201 — collapsing into 20 possible sums from 1 to 20. 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 1d20 has a plateau at 1 and 2 and 3 and 4 and 5 and 6 and 7 and 8 and 9 and 10 and 11 and 12 and 13 and 14 and 15 and 16 and 17 and 18 and 19 and 20 and why adding dice makes the distribution tighter rather than wider.