2d20 Dice Probability — Full Distribution and Odds
The basis of advantage and disadvantage — keep the higher or lower of two d20s and the curve bends toward 20 or toward 1. The sum runs from 2 to 40, the average is 21, and the most likely total is 21. 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 |
|---|---|---|---|---|---|
| 2 | 1 | 0.250% | 0.250% | 100.0% | 1 in 400 |
| 3 | 2 | 0.500% | 0.750% | 99.8% | 1 in 200 |
| 4 | 3 | 0.750% | 1.50% | 99.3% | 1 in 133 |
| 5 | 4 | 1.00% | 2.50% | 98.5% | 1 in 100 |
| 6 | 5 | 1.25% | 3.75% | 97.5% | 1 in 80 |
| 7 | 6 | 1.50% | 5.25% | 96.3% | 1 in 66.7 |
| 8 | 7 | 1.75% | 7.00% | 94.8% | 1 in 57.1 |
| 9 | 8 | 2.00% | 9.00% | 93.0% | 1 in 50 |
| 10 | 9 | 2.25% | 11.3% | 91.0% | 1 in 44.4 |
| 11 | 10 | 2.50% | 13.8% | 88.8% | 1 in 40 |
| 12 | 11 | 2.75% | 16.5% | 86.3% | 1 in 36.4 |
| 13 | 12 | 3.00% | 19.5% | 83.5% | 1 in 33.3 |
| 14 | 13 | 3.25% | 22.8% | 80.5% | 1 in 30.8 |
| 15 | 14 | 3.50% | 26.3% | 77.3% | 1 in 28.6 |
| 16 | 15 | 3.75% | 30.0% | 73.8% | 1 in 26.7 |
| 17 | 16 | 4.00% | 34.0% | 70.0% | 1 in 25 |
| 18 | 17 | 4.25% | 38.3% | 66.0% | 1 in 23.5 |
| 19 | 18 | 4.50% | 42.8% | 61.8% | 1 in 22.2 |
| 20 | 19 | 4.75% | 47.5% | 57.3% | 1 in 21.1 |
| 21 | 20 | 5.00% | 52.5% | 52.5% | 1 in 20 |
| 22 | 19 | 4.75% | 57.3% | 47.5% | 1 in 21.1 |
| 23 | 18 | 4.50% | 61.8% | 42.8% | 1 in 22.2 |
| 24 | 17 | 4.25% | 66.0% | 38.3% | 1 in 23.5 |
| 25 | 16 | 4.00% | 70.0% | 34.0% | 1 in 25 |
| 26 | 15 | 3.75% | 73.8% | 30.0% | 1 in 26.7 |
| 27 | 14 | 3.50% | 77.3% | 26.3% | 1 in 28.6 |
| 28 | 13 | 3.25% | 80.5% | 22.8% | 1 in 30.8 |
| 29 | 12 | 3.00% | 83.5% | 19.5% | 1 in 33.3 |
| 30 | 11 | 2.75% | 86.3% | 16.5% | 1 in 36.4 |
| 31 | 10 | 2.50% | 88.8% | 13.8% | 1 in 40 |
| 32 | 9 | 2.25% | 91.0% | 11.3% | 1 in 44.4 |
| 33 | 8 | 2.00% | 93.0% | 9.00% | 1 in 50 |
| 34 | 7 | 1.75% | 94.8% | 7.00% | 1 in 57.1 |
| 35 | 6 | 1.50% | 96.3% | 5.25% | 1 in 66.7 |
| 36 | 5 | 1.25% | 97.5% | 3.75% | 1 in 80 |
| 37 | 4 | 1.00% | 98.5% | 2.50% | 1 in 100 |
| 38 | 3 | 0.750% | 99.3% | 1.50% | 1 in 133 |
| 39 | 2 | 0.500% | 99.8% | 0.750% | 1 in 200 |
| 40 | 1 | 0.250% | 100.0% | 0.250% | 1 in 400 |
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 | 88.8% | 1 in 1.13 |
| ≥ 16 | 73.8% | 1 in 1.36 |
| ≥ 22 | 47.5% | 1 in 2.11 |
| ≥ 28 | 22.8% | 1 in 4.4 |
| ≥ 33 | 9.00% | 1 in 11.1 |
Reading the shape
Each die is uniform, but the sum is not. 2d20 has 400 ordered outcomes spread over 39 totals, and the counts rise toward the middle because a middle total can be assembled from more orderings. The curve is symmetric around 21 — for every way to roll 20 there is a mirrored way to roll 22 — which is why the standard deviation of 8.15 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 2d20?
21. The formula is dice × (faces + 1) / 2 — 2 × (20 + 1) / 2 — which works for any fair die. The standard deviation is 8.15, so most rolls land between 12 and 30.
What is the most likely total on 2d20?
Exactly 21, at 5.00% — 20 of the 400 equally likely outcomes. The extremes (2 and 40) take 1 way each, or 0.250%.
How many possible outcomes does 2d20 have?
400 ordered outcomes — 202 — collapsing into 39 possible sums from 2 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 2d20 has a single peak at 21 and why adding dice makes the distribution tighter rather than wider.