Dice Probability Reference: d4 to d20

Single dice: flat by definition

One die of any size gives every face an identical probability: 25% on a d4, 16.67% on a d6, 10% on a d10, 5% on a d20. There is no centre and no tail — every outcome is as likely as every other.

The practical consequence is that a single-die system has maximum variance for its range. A d20 roll needing 15 or higher succeeds 30% of the time, and the difference between needing 15 and needing 16 is exactly 5 percentage points, every time, anywhere on the scale.

That linearity is why d20 systems are easy to reason about and why a plus-one modifier is worth the same whether you needed a 5 or a 19.

Sums: the bell curve appears immediately

Add two dice and the flat distribution becomes triangular. On 2d6, a total of 7 has six ways to occur and a total of 2 has one, so 7 comes up 16.67% of the time and 2 comes up 2.78%.

Add a third and it becomes recognisably bell-shaped. On 3d6 the middle results, 10 and 11, each come up about 12.5%, while 3 and 18 each come up about 0.46% — roughly one roll in 216.

The practical effect is that a sum-based system is much more predictable than a single-die one with a similar range. A 3d6 result between 8 and 13 covers about 68% of rolls, so a character with a modifier that shifts the curve by two points changes their outcomes far more than a two-point shift on a d20 would.

• 2d6: total 7 is most likely at 16.67%; totals 2 and 12 at 2.78% each.
• 3d6: totals 10 and 11 at about 12.5%; totals 3 and 18 at about 0.46%.
• The more dice you add, the tighter the distribution around the middle.
• Modifiers matter more on a curve than on a flat distribution.

Advantage: what rolling twice is worth

Rolling two d20 and taking the higher is worth different amounts depending on what you needed. It is worth almost nothing when you needed a 2, worth almost nothing when you needed a 20, and worth the most in the middle.

The arithmetic is simple: you fail only if both dice fail, so the failure chance is squared. Needing 11 or higher, a single die succeeds 50% of the time and advantage succeeds 75% — a gain of 25 percentage points. Needing 2 or higher, a single die succeeds 95% and advantage succeeds 99.75%, a gain of under 5.

The peak is at a target of 11, and the effect is symmetric around it. Advantage is worth roughly the same as a +5 modifier in the middle of the range and roughly a +1 at either end, which is why systems that use it feel different from systems with flat bonuses.

Streaks and how often a d20 fails three times

The question that comes up at every table is whether a run of bad rolls is evidence of a bad die. Almost always it is not, and the arithmetic is easy enough to do out loud.

Failing a 50/50 roll three times running has probability 0.5 cubed, which is 12.5% — about one sequence in eight. Over a three-hour session with sixty rolls, a run of three failures somewhere is close to certain.

Failing five times running at 50/50 is about 3%, which sounds rare and will still happen to somebody at the table most sessions. Distinguishing an actually biased die from ordinary bad luck needs hundreds of recorded rolls, not a memorable evening.

Dice pools: a different question entirely

A pool system asks how many dice cleared a threshold rather than what they totalled, and the resulting distribution is binomial rather than a sum of uniforms. That makes it behave completely differently as the pool grows.

Six d10s needing 7 or higher succeed 40% each, so the expected count is 2.4. But zero successes happens about 4.7% of the time — roughly one roll in twenty-one — while a summed roll of six dice essentially never produces its minimum.

That wide spread is why pool systems feel swingy. The expected value is reassuring and the distribution is not: a system that hands out six dice at a forty percent success rate and asks for three successes fails 54% of the time, despite an expected count of 2.4.

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