The basic rule
An exploding die is one that triggers another roll when it shows its highest face. You add the new roll to the total, and if the new roll also shows the highest face, you roll again. The chain stops when a roll comes up with anything lower. Because the sequence has no fixed upper limit, such rolls are also called open-ended.
Take an exploding six-sided die. Roll a 4, and the result is 4. Roll a 6, and you keep the 6 and roll again. If the second roll is a 3, the total is 9. If it is another 6, you are at 12 and rolling a third time.
Some games explode on the top two faces, or only on a specific number, and some cap the number of times a die may explode. The logic below applies to the standard version, which re-rolls only on the maximum.
The chance of each total
Since each roll is independent, the chance of a long chain shrinks quickly. For an exploding six-sided die:
- A chain of at least one six has probability 1/6.
- At least two sixes in a row: (1/6) × (1/6) = 1/36.
- At least three sixes in a row: 1/216.
Totals follow from these. A total of 1 through 5 means the first roll stopped the chain, so each has probability 1/6. A total of 7 requires a six and then a one, so it has probability (1/6) × (1/6) = 1/36. The same is true of totals 8, 9, 10 and 11. A total of 13 requires two sixes and then a one, giving (1/6)^3 = 1/216.
Notice what is missing. A total of 6 is impossible, since a six always triggers another roll that adds at least one. The same goes for 12, 18 and every other multiple of six. The outcomes have gaps, which surprises people the first time they see it.
Add up all the pieces as a check. The chance of a total of 5 or less is 5/6. The chance of a total above 6 is 1/6, which is the chance of the first roll being a six. That sums to one.
The average goes up
Because of the extra rolls, an exploding die averages more than an ordinary one. Call the average E, and split the roll into two cases:
- With probability 5/6 the roll is a 1 through 5, averaging 3.
- With probability 1/6 the roll is a 6, after which you get another full exploding roll, worth E on average.
So E = (5/6) × 3 + (1/6) × (6 + E). That is E = 2.5 + 1 + E/6, so E × (5/6) = 3.5, which gives E = 4.2.
An ordinary six-sided die averages 3.5, so the explosion adds 0.7 to the expected value. That is a modest change compared with how dramatic the occasional huge roll feels. The extra average comes from rare events, not from every roll.
Why the tail matters
The long right-hand side of the distribution is the whole point. The chance of beating a target falls off smoothly rather than hitting a wall:
- A plain six-sided die can never beat a target of 7.
- An exploding six-sided die reaches 7 or more with probability 1/6.
- Reaching 10 or more requires a six first and then a 4 or higher on what follows, so (1/6) × (3/6) = 1/12, about 8.3 percent.
- Reaching 13 or more needs two sixes in a row, 1/36, about 2.8 percent.
This means no difficulty is truly impossible. That is why open-ended rolls are used in games that want heroic, once-in-a-while feats. A weak character can still pull off something unlikely, and a strong one can fail, but the chance of a spectacular result is not zero.
The cost is predictability. Even though the average is only slightly higher, the spread is larger, so the standard deviation of an exploding die is higher than that of an ordinary one. A designer who wants rolls to stay in a narrow band may avoid them, while one who wants tension and occasional swings may welcome them.
Exploding dice in a pool
The rule can also be applied to each die in games built around dice pools, counting a success on every roll that meets the target. This creates a different kind of tail. A pool in which sixes explode and a roll of 5 or 6 counts as success gives extra chances to succeed on the same die. It makes large success counts reachable without needing many dice, and it makes the exact odds harder to work out by hand.
For complicated combinations, it is often easier to write a short simulation than to derive a formula. A spreadsheet simulation can model a few thousand rolls and show the distribution’s shape quickly. The simulated figures will be close to the exact ones but not identical, so check them against a simple case like the 4.2 average above before trusting a more complicated model.
Practical cautions
A few points matter in play:
- Agree on the rule first. Explode on the maximum only, or on more faces? With a cap or without? The odds depend on it.
- Do not expect a “due” explosion. Each roll is independent, so a long stretch without a six does not make one more likely next time.
- Keep records of long chains. They are rare enough that players tend to remember them and over-estimate how often they occur.
Open-ended rolls are a way of adding drama, not a way of changing the underlying odds in anyone’s favour. They shift probability from the middle of the range toward the extreme and leave the rolls independent.