Closest CallExploding dice odds
Roll again and add whenever the die shows its highest face. Here is what that does to the odds.
One exploding six-sided die
An exploding die is rolled again every time it shows its highest face, and the new roll is added to the total. A six followed by a four counts as 10. There is no upper limit, though high totals get rare quickly.
| Total | Exploding d6 | Ordinary d6 |
|---|---|---|
| 2 or more | 83.3% | 83.3% |
| 3 or more | 66.7% | 66.7% |
| 4 or more | 50.0% | 50.0% |
| 5 or more | 33.3% | 33.3% |
| 6 or more | 16.7% | 16.7% |
| 7 or more | 16.7% | 0% |
| 8 or more | 13.9% | 0% |
| 9 or more | 11.1% | 0% |
| 10 or more | 8.33% | 0% |
| 11 or more | 5.56% | 0% |
| 12 or more | 2.78% | 0% |
| 13 or more | 2.78% | 0% |
| 14 or more | 2.31% | 0% |
| 15 or more | 1.85% | 0% |
| 16 or more | 1.39% | 0% |
| 17 or more | 0.93% | 0% |
| 18 or more | 0.46% | 0% |
| 19 or more | 0.46% | 0% |
| 20 or more | 0.39% | 0% |
What the chart tells you
- The average roll is 4.2, compared with 3.5 for an ordinary die. Exploding adds less than most people expect.
- You can never roll exactly 6, 12 or 18. A six always rolls again and adds at least 1, which is why "6 or more" and "7 or more" have the same chance.
- The die explodes at least once on 16.7% of rolls, twice in a row on 2.78%, and three times in a row on 0.46% (1 in 216).
- A total of 10 or more comes up 8.33% of the time, and 13 or more 2.78%.
Average roll for each die size
Smaller dice explode more often, so they gain the most.
| Die | Ordinary average | Exploding average | Chance it explodes |
|---|---|---|---|
| d4 | 2.5 | 3.33 | 25.0% |
| d6 | 3.5 | 4.20 | 16.7% |
| d8 | 4.5 | 5.14 | 12.5% |
| d10 | 5.5 | 6.11 | 10.0% |
| d12 | 6.5 | 7.09 | 8.33% |
| d20 | 10.5 | 11.05 | 5.00% |
Two exploding six-sided dice added together
| Total | Two exploding d6 | Two ordinary d6 |
|---|---|---|
| 7 or more | 58.3% | 58.3% |
| 8 or more | 47.2% | 41.7% |
| 9 or more | 38.0% | 27.8% |
| 10 or more | 30.6% | 16.7% |
| 11 or more | 25.0% | 8.33% |
| 12 or more | 21.3% | 2.78% |
| 13 or more | 16.7% | 0% |
| 15 or more | 10.0% | 0% |
| 18 or more | 5.09% | 0% |
| 20 or more | 3.01% | 0% |
Two exploding dice average 8.4, against 7 for two ordinary dice.
How to work it out
To reach a total between 7 and 11 you need a six and then the right second roll. A total of exactly 9 is a six followed by a three: 1/6 × 1/6, or 2.78%. For the average, an exploding die is worth its ordinary average multiplied by s/(s-1), where s is the number of sides: for a d6 that is 3.5 × 6/5 = 4.2.
Where this matters
Exploding dice appear in many tabletop role-playing and skirmish games, usually for damage or for heroic effort, because they make a spectacular result possible on any roll without changing the typical one very much.
More dice odds
- Dice odds library: every chart in one place
- Two dice (2d6) odds chart
- Three dice (3d6) odds chart
- 4d6 drop the lowest: ability score odds
- d20 with advantage and disadvantage
- Five of a kind and other five-dice odds
- Attacker against defender dice odds
- Two twenty-sided dice (2d20) odds chart
- At least one six in 1 to 10 rolls
- Dice pools: odds of successes on 5 or 6
- Dice calculator: the exact odds for any total with any number of dice, plus a dice roller.
Every figure on this page was computed by counting all possible outcomes. Game names mentioned here are trademarks of their owners; this site is not affiliated with or endorsed by them.