2d6 and 3d6 Probability Tables
On 2d6 there are 36 equally likely combinations, and 7 is the most common total at 16.67% (6 ways), while 2 and 12 are 2.78% each. On 3d6 there are 216 combinations: 10 and 11 tie as most likely at 12.5% each, and 3 or 18 is only 0.46%. Both average to the middle: 7 and 10.5.
How do you calculate 2d6 probability?
Each six-sided die has 6 faces, so two dice have 6 × 6 = 36 possible combinations, and every one of them is equally likely. To find the chance of a total, count the combinations that make it and divide by 36.
A total of 4 can be made three ways: 1+3, 2+2 and 3+1. So the chance of rolling exactly 4 is 3/36, or 8.33%. Note that 1+3 and 3+1 count separately, because the two dice are different objects even if they look the same.
Every number below was counted exactly, not simulated. To check any other roll, enter it in the dice probability calculator, or roll it yourself with the 2d6 roller.
What are the odds for each total on 2d6?
"At least" is the chance of rolling that number or higher, which is what you want when you need to meet a target. "At most" is that number or lower.
| Total | Ways (of 36) | Exactly | At least | At most |
|---|---|---|---|---|
| 2 | 1 | 2.78% | 100.00% | 2.78% |
| 3 | 2 | 5.56% | 97.22% | 8.33% |
| 4 | 3 | 8.33% | 91.67% | 16.67% |
| 5 | 4 | 11.11% | 83.33% | 27.78% |
| 6 | 5 | 13.89% | 72.22% | 41.67% |
| 7 | 6 | 16.67% | 58.33% | 58.33% |
| 8 | 5 | 13.89% | 41.67% | 72.22% |
| 9 | 4 | 11.11% | 27.78% | 83.33% |
| 10 | 3 | 8.33% | 16.67% | 91.67% |
| 11 | 2 | 5.56% | 8.33% | 97.22% |
| 12 | 1 | 2.78% | 2.78% | 100.00% |
- Average: 7.
- Most likely total: 7, at 16.67% (1 in 6).
- Rolling 6, 7 or 8: 44.44%, almost half of all rolls.
- Doubles: 6 of 36 combinations, or 16.67%.
What are the odds for each total on 3d6?
Three dice have 6 × 6 × 6 = 216 combinations. The counting works the same way, there are just more of them.
| Total | Ways (of 216) | Exactly | At least | At most |
|---|---|---|---|---|
| 3 | 1 | 0.46% | 100.00% | 0.46% |
| 4 | 3 | 1.39% | 99.54% | 1.85% |
| 5 | 6 | 2.78% | 98.15% | 4.63% |
| 6 | 10 | 4.63% | 95.37% | 9.26% |
| 7 | 15 | 6.94% | 90.74% | 16.20% |
| 8 | 21 | 9.72% | 83.80% | 25.93% |
| 9 | 25 | 11.57% | 74.07% | 37.50% |
| 10 | 27 | 12.50% | 62.50% | 50.00% |
| 11 | 27 | 12.50% | 50.00% | 62.50% |
| 12 | 25 | 11.57% | 37.50% | 74.07% |
| 13 | 21 | 9.72% | 25.93% | 83.80% |
| 14 | 15 | 6.94% | 16.20% | 90.74% |
| 15 | 10 | 4.63% | 9.26% | 95.37% |
| 16 | 6 | 2.78% | 4.63% | 98.15% |
| 17 | 3 | 1.39% | 1.85% | 99.54% |
| 18 | 1 | 0.46% | 0.46% | 100.00% |
- Average: 10.5.
- Most likely totals: 10 and 11, tied at 12.5% each (27 ways).
- Rolling 9 to 12: 48.15%, nearly half of all rolls.
- Rolling 18: 0.46%, or 1 in 216. Rolling a 3 is just as rare.
- Rolling 15 or higher: 9.26%, a little under 1 in 10.
You can roll it with the 3d6 roller. 3d6 is the old-school method for ability scores; the modern default, 4d6 drop lowest, is covered in how to roll stats in 5e.
Why do 2d6 and 3d6 make a bell curve?
A single die is flat: each face has the same 1 in 6 chance. Adding dice together changes that, because middle totals can be built in many more ways than extreme ones.
On 2d6 there is only one way to roll 12 (6 and 6), but six ways to roll 7: 1+6, 2+5, 3+4, 4+3, 5+2 and 6+1. Whatever the first die shows, there is always exactly one face on the second die that brings the total to 7. No other total has that property, which is why 7 sits at the peak.
With two dice the result is a triangle that rises steadily to 7 and falls away. With three dice the counts become 1, 3, 6, 10, 15, 21, 25, 27, then back down, which is a smoother, rounder bell. Each extra die makes the curve smoother and pulls more of the results toward the middle.
How spread out are the results?
One way to measure spread is the standard deviation. For 2d6 it is about 2.42, and for 3d6 about 2.96. Relative to the full range, though, 3d6 is tighter: its middle four totals (9 to 12) cover 48.15% of rolls, while its extremes are under half a percent each.
How do you use these tables at the table?
Most of the time you want to know your chance of reaching a target, so read the "At least" column. Some quick examples:
- Need 8 or more on 2d6: 41.67%. Need 7 or more and it jumps to 58.33%. One point near the middle of a bell curve moves the odds a lot.
- Need 10 or more on 2d6: 16.67%, the same as rolling exactly 7.
- Need 15 or more on 3d6: 9.26%. Need 16 or more and it halves to 4.63%.
- Want to stay at 5 or lower on 3d6: 4.63%, the mirror image of 16 or higher.
Notice that on a flat d20, every extra point you need costs the same 5%. On 2d6 or 3d6, a point costs a lot near the middle and very little at the edges. That's the main practical difference between a single die and a sum of dice.
What are the odds in an opposed 2d6 or 3d6 roll?
When two players each roll 2d6 and compare, a tie happens 11.27% of the time, and each side wins outright 44.37% of the time. With 3d6 against 3d6, ties drop to 9.28% and each side wins 45.36%.
What happens with even more dice?
The pattern keeps going. 4d6 averages 14 and its most likely total, 14, comes up 11.27% of the time, while rolling the maximum 24 is about 0.08%. A Fireball's 8d6 averages 28, and 64.22% of all Fireballs land between 24 and 32. The more dice you add, the more reliably the total lands near the average, which is why big damage rolls rarely feel wildly swingy.
How is 2d6 different from 1d12?
Both can roll 12, but they behave very differently. A d12 is flat: every number from 1 to 12 has an 8.33% chance, and the average is 6.5. 2d6 can't roll a 1, averages 7, and lands on 7 twice as often as a d12 lands on any single number. In D&D both a greatsword (2d6) and a greataxe (1d12) share the same maximum, but the greatsword is more consistent and averages half a point more.
| Roll | Range | Average | Chance of max | Shape |
|---|---|---|---|---|
| 1d12 | 1–12 | 6.5 | 8.33% | Flat |
| 2d6 | 2–12 | 7 | 2.78% | Triangle, peak at 7 |
| 1d20 | 1–20 | 10.5 | 5% | Flat |
| 3d6 | 3–18 | 10.5 | 0.46% | Bell, peak at 10–11 |
Where are 2d6 and 3d6 used in games?
- Weapon damage: the greatsword and maul deal 2d6. More examples are on the average dice rolls cheat sheet.
- Ability scores: classic D&D rolled 3d6 for each score.
- Other systems: many games use 2d6 or 3d6 as their main resolution roll because the bell curve makes average results common and extreme ones rare, unlike the flat d20.
FAQ
What is the most common roll on 2d6?
7. Six of the 36 combinations add up to 7 (1+6, 2+5, 3+4 and the reverses), so it comes up 16.67% of the time, about once every six rolls.
What are the odds of rolling 12 on 2d6?
1 in 36, or 2.78%. Only one combination, a 6 on both dice, makes 12. Rolling a 2 (double 1s) has the same odds.
What are the odds of rolling 18 on 3d6?
1 in 216, or 0.46%. You need a 6 on all three dice.
What is the average of 3d6?
10.5. Each d6 averages 3.5, and three of them average 3 times 3.5. The totals 10 and 11 are tied as the most likely results.
What are the odds of rolling doubles on 2d6?
6 in 36, or 16.67%. There are six doubles (1-1 through 6-6) out of 36 possible combinations.
Related
- Dice Probability Calculator
- Roll 2d6
- Roll 3d6
- Average Dice Rolls: d4 to d100 Cheat Sheet
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Contains content from the System Reference Document 5.1 and 5.2 by Wizards of the Coast LLC, used under the Creative Commons Attribution 4.0 International License.