Dice probability in tabletop RPGs: advantage, disadvantage and exploding dice

The d20 is a line, not a bell

A d20 has 20 faces and each one comes up with the same odds: 1/20, or 5%. That is a uniform distribution, also called "flat," because if you plot a bar chart of each outcome’s chance, every bar is the same height. There is no "most common result" on a lone d20, a 10 shows up exactly as often as a 1 or a 20. That gives a simple formula: the chance a single d20 beats a minimum result n is (21 − n)/20, because there are (21 − n) faces from n up to 20. Needing an 11 or higher, for example, covers results 11 through 20, ten faces out of twenty, exactly 50%.

That flat baseline is what three classic RPG mechanics each bend in a different direction. D&D’s advantage and disadvantage roll the same d20 twice and pick which half to keep. Systems like GURPS swap the d20 for 3d6, replacing the line with a bell. And "exploding" dice let a single, usually small, die keep growing whenever it hits the ceiling. The next three sections derive the math for each, worked by hand and checked by simulation, not copied from some forum thread.

Uniform distribution

Every outcome has the same probability. A lone d20 is the classic example: 5% per face.

Bell curve

Middle results come up far more than the extremes, because there are more dice combinations that sum to an average value.

Difficulty (DC)

The minimum result a check requires. "Needs 11+" is a 50% difficulty on a normal d20.

Exploding die

On rolling the maximum face, you roll again and add it, potentially chaining several times in a row.

Expected value

The long-run average of a roll, weighting each possible outcome by its probability.

Advantage and disadvantage: the math of rolling twice

The rule, taken straight from the D&D 5th edition System Reference Document (SRD), is simple to state: when you have advantage, roll two d20s and use the higher result; when you have disadvantage, roll two and use the lower. That is exactly what the dice roller does in its advantage/disadvantage mode: it rolls both throws and shows which one counts. The part that trips people up is assuming "roll it twice" doubles your odds. It does not, what changes is the probability that at least one of the two throws beats the target, and that math depends on the difficulty.

1. Start with a single die’s chanceOn a normal d20, (21 − n)/20 of the results beat n or higher. For n = 11, that is 10/20 = 50%.
2. Advantage is the complement of "both fail"With advantage, you only fail if BOTH d20s roll below n. One die’s chance of failing is (n − 1)/20; since the two are independent, the chance both fail is that fraction squared.
3. Subtract from 1P(advantage ≥ 11) = 1 − (10/20)² = 1 − 0.25 = 0.75, i.e. 75%. Check it: with two d20s, only 100 of the 400 possible pairs (20 × 20) have both throws below 11, and 400 − 100 = 300, exactly 75% of 400.
4. Disadvantage is the mirrorWith disadvantage, you only succeed if BOTH dice beat n. P(disadvantage ≥ 11) = (10/20)² = 0.25, i.e. 25%, the same amount advantage gained, just on the bad side.
5. Check the extremesAt n = 1 (impossible to fail) and n = 20 (only a natural 20 works), advantage and disadvantage sit almost on top of the normal line: rolling twice barely helps when the result was already near-certain or near-impossible.

The effect of advantage across every difficulty, 1 through 20

Plotting P(≥ n) for the three curves, normal, advantage and disadvantage, across every difficulty from 1 to 20 shows where advantage actually pays off. The gain peaks exactly where the odds were already a coin flip: needing 11+ gains 25 percentage points (from 50% to 75%). At both extremes, needing 2+ or 20+, the gain shrinks to 4.75 percentage points, because there is little room to improve a result that was already near-certain, and little to gain from one that was already near-impossible. The curve is symmetric: the same pattern mirrors for disadvantage.

The practical lesson for a game master: granting advantage on a check that was already easy (needing 6+) is a small treat, it rises from 75% to 93.75%; granting it on a tight check (needing 16+) is what actually changes the table, nearly doubling the odds, from 25% to 43.75%. That is why advantage "feels" stronger on hard checks: in relative terms, it genuinely is.

Adding dice creates a bell: 3d6 versus the d20

Not every system resolves checks with a lone d20. GURPS, from Steve Jackson Games, uses 3d6 as its base mechanic: you roll three six-sided dice, add them up, and need to roll equal to or under your skill level ("roll under"). Traveller, in its current form published by Mongoose, uses a related but distinct variant with two dice: roll 2d6, add modifiers, and beat a target number, usually 8. Both trade the d20’s flat line for a curve, because adding independent dice concentrates the outcome toward the middle: there are far more ways to sum to a middling value than to an extreme one.

You can see this by counting combinations. In 3d6 there are 6³ = 216 possible ordered outcomes (each die, 6 values, three dice). Only one single combination sums to 3 (1-1-1) and only one sums to 18 (6-6-6). But summing to 10, for instance, has six different value sets: {1,3,6}, {1,4,5}, {2,2,6}, {2,3,5}, {2,4,4} and {3,3,4}, and each one can land in a different order across the three dice. Counting the orderings (permutations) of each set gives exactly 27 ways to roll a 10, and another 27 to roll an 11. In other words, rolling a 10 or 11 in 3d6 is 27 times more likely than rolling a 3 or an 18.

In practice, this changes the system’s "feel." On a d20, an average character and an exceptional one have equally wide curves: a +3 bonus shifts the whole line evenly, at any difficulty. In a 3d6 system, that same +3 bonus shifts the curve, but the effect is stronger near the center (where most checks land) and weaker at the tails, because there is little probability mass left there to shift. It is the same logic as D&D’s advantage, just baked into the shape of the dice instead of into a separate rule.

Exploding dice: the geometric series hiding in a d6

Some systems use "open-ended" or "exploding" dice: rolling the maximum face lets you roll again and add the result, potentially chaining indefinitely. Savage Worlds, from Pinnacle Entertainment Group, calls this an "Ace" (a die that "aces"), and applies it to every trait and damage roll. On top of that, players and important NPCs ("Wild Cards") roll an extra six-sided die called the Wild Die alongside their trait die, keeping the higher of the two, and the Wild Die can ace as well. An exploding die can, in theory, keep growing forever, but each extra explosion is geometrically rarer than the last, so the expected value still comes out to a finite, calculable number.

1. The chance of exploding k times in a rowEach explosion requires rolling the maximum, a 1/n chance on an n-sided die. Exploding k times in a row has chance (1/n)ᵏ, rarer with every step.
2. Sum that geometric seriesThe expected number of dice rolled before stopping is 1 + 1/n + 1/n² + ... , a classic geometric series that converges to 1/(1 − 1/n) = n/(n − 1).
3. Multiply by an ordinary die’s averageEach individual die rolled, whether it explodes or not, still averages (n + 1)/2. The total expected value is that average times the expected number of dice.
4. Close the formulaE = (n + 1)/2 × n/(n − 1) = n(n + 1) / (2(n − 1)).

Checking it on a d6: E = 6 × 7 / (2 × 5) = 42/10 = 4.2, against the normal average of (6 + 1)/2 = 3.5, a 20% increase. I ran a 2-million-roll simulation of an exploding d6 and the mean came out to 4.2016, nearly identical to the closed-form value. Smaller dice explode proportionally more: an exploding d4 (a 1/4 chance each roll) has its average lifted by 33.3%, while an exploding d12 (only a 1/12 chance) rises just 9.1%. That is why Savage Worlds’ Wild Die uses a d6 specifically, small enough to visibly explode at the table.

What this changes at the table

All three mechanics solve the same problem, injecting controlled variance into a check, in different ways. Advantage and disadvantage are a punctual adjustment: they weigh most on checks that were already uncertain and barely move the extremes, which makes them predictable to apply even without doing any math on the spot. Swapping the d20 for 3d6 (or 2d6, as in Traveller) is a structural change: it lowers the chance of extreme results on every check, all the time, so average characters rarely "botch it badly" and exceptional ones rarely "just get lucky." And exploding dice keep a long tail permanently alive: however small the odds, a huge result is never fully out of reach, which is great for the climax of a scene, but means accepting that, once in a while, the dice will run away from you.

The same approach, breaking a game mechanic down to a first-principles calculation, shows up elsewhere in the catalog: gacha game pity systems use an accumulating-probability logic similar to exploding dice, and poker hand probability is another case of counting combinations to see why certain results are so much rarer than others.

• Is the check’s difficulty near 50%? That is where advantage and disadvantage weigh the most.
• Swapping a d20 for summed dice (like 3d6)? Expect fewer extreme results, not more drama.
• Can the die explode? The fewer sides it has, the more often it tends to ace.
• Did you test the math on small numbers before applying it to the whole table? It is worth simulating in the dice roller first.