Exploding Dice Probability Calculator
Enter a pool of standard additive exploding dice and a target total. The result gives the exact chance to reach it, rather than treating an explosion as only one extra roll.
Chance to meet or beat target30.56%
- Chance to roll exactly the target
- 5.56%
- Average final total
- 8.4
- Standard deviation
- 4.61
- Median total
- 7
- 90th-percentile total
- 14
- Chance of at least one explosion
- 30.56%
- Expected extra rolls
- 0.4
Final-total probabilities near the target
| Final total | Exactly this total | This total or higher |
|---|---|---|
| 7 | 11.11% | 58.33% |
| 8 | 9.26% | 47.22% |
| 9 | 7.41% | 37.96% |
| 10 | 5.56% | 30.56% |
| 11 | 3.7% | 25% |
| 12 | 4.63% | 21.3% |
| 13 | 3.7% | 16.67% |
Assumes fair, independent dice; a maximum face is added and rolls the same die again without a cap.
The infinite explosion tail is retained until its remaining probability is below 0.000000000001.
How to use this calculator
- Enter how many identical dice are rolled and added together.
- Enter the number of sides on each die; only its highest face explodes.
- Add any fixed roll modifier, including a negative penalty.
- Enter the total you need, then compare the target chance with the nearby-total table.
How exploding dice totals are calculated
With a standard additive explosion, a die keeps its maximum face and immediately rolls again. A d6 that shows 6 then 3 therefore totals 9; 6, 6, then 2 totals 14. The added rolls can also explode, so there is no fixed highest result.
Let N be the number of dice, S the sides on each die, B the modifier, and Q the target. One exploding die can finish at kS + v, where k is the number of maximum faces in a row and v is the final face from 1 through S − 1. Its probability is P(Y = kS + v) = 1 / S^(k + 1).
The calculator convolves that distribution for all N dice, then applies the modifier: T = Y₁ + Y₂ + … + Yₙ + B. The headline is P(T ≥ Q), the sum of every final total at or above your target. The exact-result row is P(T = Q). The nearby-total table makes the same summed distribution visible around the decision point.
Average, spread, and explosions
The exact mean of one exploding dS is S(S + 1) / [2(S − 1)], so the average final total is N × mean + B. Its variance is S(S + 1)(S² + 7S − 2) / [12(S − 1)²]; the displayed standard deviation is the square root of N times that variance. Median and 90th-percentile totals come from the cumulative discrete distribution, not by rounding the average.
For a d6, the average of one ordinary die is 3.5, while one exploding d6 averages 4.2. Smaller dice have a heavier upper tail because a maximum comes up more often. More dice make middle totals more common, but a single long explosion chain can still create a high result.
What this calculator leaves out
This is the standard rule where the maximum face is counted before the reroll. It does not cover success-counting pools, penetrating dice, capped explosions, reroll-and-replace rules, several exploding faces, Wild Dice, keep-highest rules, critical effects, or a game-specific interpretation of a modifier. The infinite tail is cut only after the unallocated probability is below one trillionth, so extremely remote totals may display as zero.
Worked example
Suppose you roll two exploding d6s with no modifier and need 10 or more. The chance is 11 / 36, or 30.56%. It includes the ordinary 5 + 5 result, one explosion followed by enough on the other terminating faces, and every larger explosion chain.
The chance of exactly 10 is 1 / 18, or 5.56%, and the average final total is 8.4. The median is 7, while the 90th-percentile total is 14, which shows why the average alone does not describe the upper tail.
Common questions
What does it mean when a die explodes?
An exploding die rolls again when it shows its maximum face, and both face values are added. If that new roll is also the maximum, it rolls again. This calculator uses that unlimited, additive version of the rule.
Can an exploding die keep rolling forever?
In principle, yes: every maximum face starts another roll, so there is no highest possible total. In practice, each additional maximum is less likely by another factor of 1 divided by the die size. The calculation keeps outcomes until the remaining tail is below one trillionth.
Why can a single exploding d6 never total exactly 6 or 12?
A 6 does not stop the die; it adds 6 and requires another roll. A one-explosion d6 ends at 7 through 11, and a two-explosion d6 ends at 13 through 17. Positive multiples of 6 are therefore impossible for one standard exploding d6, although they can occur when several dice are added.
How do multiple exploding dice change the probability curve?
Adding dice combines their totals, so middle results have many more combinations than very low results. Explosions retain a long upper tail because any starting die can keep adding maximum faces. The tool convolves the dice distributions rather than approximating the pool with a normal curve.
What is the average result of an exploding d6?
One standard exploding d6 has an average of 4.2, compared with 3.5 for an ordinary d6. The extra 0.7 comes from the chance of rolling a 6 and receiving another full d6 roll. This is a long-run average, not the most likely result.
How is this different from counting successes in a dice pool?
This calculator adds face values into one total and compares that total with a target. A success-counting pool instead counts how many individual dice clear a threshold, usually ignoring their exact face values beyond success or failure. The two systems have different probability distributions.