The bell curve
One die is flat: every face is as likely as the next. Add a third die and the totals pile up in the middle. Three dice make 216 combinations, and 54 of them add up to 10 or 11, one roll in four. A 3 or an 18 needs all three dice to match, once in 216 rolls each.
| Total | Chance of exactly | Chance of this or more |
|---|---|---|
| 3 | 0.5% | 100% |
| 4 | 1.4% | 99.5% |
| 5 | 2.8% | 98.1% |
| 6 | 4.6% | 95.4% |
| 7 | 6.9% | 90.7% |
| 8 | 9.7% | 83.8% |
| 9 | 11.6% | 74.1% |
| 10 | 12.5% | 62.5% |
| 11 | 12.5% | 50.0% |
| 12 | 11.6% | 37.5% |
| 13 | 9.7% | 25.9% |
| 14 | 6.9% | 16.2% |
| 15 | 4.6% | 9.3% |
| 16 | 2.8% | 4.6% |
| 17 | 1.4% | 1.9% |
| 18 | 0.5% | 0.5% |
The curve is mirrored around 10.5: a 7 is as likely as a 14, and a 3 as rare as an 18.
Old-school ability scores
Before "roll four, drop the lowest" took over, characters were made with 3d6 in order: Strength, then Dexterity, and on down the list, no swapping. It makes heroes who look like ordinary people, which was the point. A 15 or better comes up 9.3% of the time, so more than half of all 3d6 characters (55.8%) have no score above 14. The modern method has its own roller, and it lifts the average from 10.5 to 12.24.
Rolling under your skill
GURPS and other roll-under games use 3d6 against a skill number, and the curve makes every point count most in the middle:
| Skill | Chance to roll it or under |
|---|---|
| 8 | 25.9% |
| 10 | 50.0% |
| 11 | 62.5% |
| 12 | 74.1% |
| 13 | 83.8% |
| 14 | 90.7% |
| 16 | 98.1% |
Going from 10 to 11 adds 12.5 points of success; going from 14 to 15 adds 4.6.
3d6 damage in D&D
- Burning Hands. The 1st-level spell deals 3d6 fire in a 15-foot cone, 10.5 on average.
- Vampiric Touch. 3d6 necrotic, and you heal half of it.
- Sneak Attack. A rogue's extra damage reaches 3d6 at 5th level.
For 3d6 plus a modifier, or mixed rolls, use the dice calculator. One die at a time lives on the d6 page; two dice on the 2d6 roller, four on the 4d6 roller.
Spells that roll 3d6
Base damage in the SRD, before upcasting or cantrip levels.
- Burning Hands (1st level)
- Vampiric Touch (3rd level)
- Black Tentacles (4th level)
- Dream (5th level)
Questions
What does 3d6 mean?
Three six-sided dice, added together. The first number is how many dice, the second how many sides, so 3d6 totals anywhere from 3 to 18. The Dice Notation Guide covers the rest.
What is the average of 3d6?
Ten and a half. Each d6 averages 3.5, and three of them make 10.5. The two most likely totals, 10 and 11, come up 12.5% of the time each.
What are the odds of rolling 18 on 3d6?
One in 216, about 0.46%: all three dice must show a 6. A 17 or more comes up 1.9% of the time, and a 15 or more 9.3%.
What are the odds of at least one 6 on three dice?
About 42.1%. The chance of no 6 on any of the three is (5/6) cubed, 57.9%.
What are the odds of rolling triples on 3d6?
One in 36, about 2.8%. Six of the 216 combinations are three of a kind. All three dice show different faces 55.6% of the time.
Is this 3d6 roller fair?
Yes. Each die comes from your browser's cryptographic random number generator with no bias towards any face, and every die is shown before it is added.