1. Two Six-Sided Dice (2d6) Probability
The 2d6 distribution is the foundation of classics like Settlers of Catan, Craps, and Powered by the Apocalypse RPGs. There are 36 possible combination outcomes ranging from 2 to 12.
Probability Distribution (2d6)
Exact mathematical probability for each sum outcome.
| Sum | Combinations | Exact Odds | At Least (≥) | At Most (≤) |
|---|---|---|---|---|
| 2 | 1 / 36 | 2.78% | 100% | 2.78% |
| 3 | 2 / 36 | 5.56% | 97.22% | 8.33% |
| 4 | 3 / 36 | 8.33% | 91.67% | 16.67% |
| 5 | 4 / 36 | 11.11% | 83.33% | 27.78% |
| 6 | 5 / 36 | 13.89% | 72.22% | 41.67% |
| 7 | 6 / 36 | 16.67% | 58.33% | 58.33% |
| 8 | 5 / 36 | 13.89% | 41.67% | 72.22% |
| 9 | 4 / 36 | 11.11% | 27.78% | 83.33% |
| 10 | 3 / 36 | 8.33% | 16.67% | 91.67% |
| 11 | 2 / 36 | 5.56% | 8.33% | 97.22% |
| 12 | 1 / 36 | 2.78% | 2.78% | 100% |
2. Three Six-Sided Dice (3d6) Probability
Used in systems like GURPS and Hero System, 3d6 yields 216 total combinations ranging from 3 to 18 with a pronounced bell curve peaking at 10.5.
Probability Distribution (3d6)
Exact mathematical probability for each sum outcome.
| Sum | Combinations | Exact Odds | At Least (≥) | At Most (≤) |
|---|---|---|---|---|
| 3 | 1 / 216 | 0.46% | 100% | 0.46% |
| 4 | 3 / 216 | 1.39% | 99.54% | 1.85% |
| 5 | 6 / 216 | 2.78% | 98.15% | 4.63% |
| 6 | 10 / 216 | 4.63% | 95.37% | 9.26% |
| 7 | 15 / 216 | 6.94% | 90.74% | 16.2% |
| 8 | 21 / 216 | 9.72% | 83.8% | 25.93% |
| 9 | 25 / 216 | 11.57% | 74.07% | 37.5% |
| 10 | 27 / 216 | 12.5% | 62.5% | 50% |
| 11 | 27 / 216 | 12.5% | 50% | 62.5% |
| 12 | 25 / 216 | 11.57% | 37.5% | 74.07% |
| 13 | 21 / 216 | 9.72% | 25.93% | 83.8% |
| 14 | 15 / 216 | 6.94% | 16.2% | 90.74% |
| 15 | 10 / 216 | 4.63% | 9.26% | 95.37% |
| 16 | 6 / 216 | 2.78% | 4.63% | 98.15% |
| 17 | 3 / 216 | 1.39% | 1.85% | 99.54% |
| 18 | 1 / 216 | 0.46% | 0.46% | 100% |
3. Single 20-Sided Die (1d20) Distribution
A single d20 has a flat uniform distribution: each face from 1 to 20 has an equal 5.0% probability.
Probability Distribution (1d20)
Exact mathematical probability for each sum outcome.
| Sum | Combinations | Exact Odds | At Least (≥) | At Most (≤) |
|---|---|---|---|---|
| 1 | 1 / 20 | 5% | 100% | 5% |
| 2 | 1 / 20 | 5% | 95% | 10% |
| 3 | 1 / 20 | 5% | 90% | 15% |
| 4 | 1 / 20 | 5% | 85% | 20% |
| 5 | 1 / 20 | 5% | 80% | 25% |
| 6 | 1 / 20 | 5% | 75% | 30% |
| 7 | 1 / 20 | 5% | 70% | 35% |
| 8 | 1 / 20 | 5% | 65% | 40% |
| 9 | 1 / 20 | 5% | 60% | 45% |
| 10 | 1 / 20 | 5% | 55% | 50% |
| 11 | 1 / 20 | 5% | 50% | 55% |
| 12 | 1 / 20 | 5% | 45% | 60% |
| 13 | 1 / 20 | 5% | 40% | 65% |
| 14 | 1 / 20 | 5% | 35% | 70% |
| 15 | 1 / 20 | 5% | 30% | 75% |
| 16 | 1 / 20 | 5% | 25% | 80% |
| 17 | 1 / 20 | 5% | 20% | 85% |
| 18 | 1 / 20 | 5% | 15% | 90% |
| 19 | 1 / 20 | 5% | 10% | 95% |
| 20 | 1 / 20 | 5% | 5% | 100% |
Frequently Asked Questions
Why does 2d6 have a bell curve while 1d20 is flat? ↓
A single die (like 1d20) has a uniform distribution where every individual face has an equal 1 in 20 chance (5%). When rolling two or more dice (like 2d6), you sum multiple independent random variables. Multiple different combinations can yield a sum of 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1), but only a single combination yields 2 (1+1) or 12 (6+6). This creates a central peak known as the Central Limit Theorem.
How do you calculate the odds of rolling at least a specific target number? ↓
Cumulative probability ("at least" or ≥ X) sums the probabilities of every outcome equal to or greater than X. For example, rolling at least a 15 on a d20 means summing outcomes 15, 16, 17, 18, 19, and 20 (6 outcomes × 5% = 30%).
What is the standard deviation in dice rolling? ↓
Standard deviation measures how widely the roll outcomes spread around the average (mean). A low standard deviation means rolls cluster tightly near the center; a high standard deviation means wider variation and greater swinginess in gameplay.