D20 Probability for D&D and TRPGs

Why the D20 became standard

Each face of a D20 carries exactly five percent. Twenty divides cleanly into a hundred, which makes probability arithmetic trivial. Raise a target number by one and the success rate drops by precisely five percentage points.

That property is decisive for rule design. Both designers and players can immediately see what a single point of bonus contributes. A D6 moves in steps of 16.7 percent, which is far too coarse, while a D100 moves in one percent steps, which is finer than anyone needs.

The icosahedron also has the most faces of any regular polyhedron, meaning it is the most granular shape that is still physically uniform.

Target numbers and success rates

For a check where you succeed on a target number or higher, just count the faces that succeed. A target of 11 succeeds on 11 through 20, which is ten faces, so fifty percent.

Across typical difficulties: target 10 gives fifty-five percent, target 15 gives thirty percent, and target 20 gives five percent. With a plus five bonus, a target of 15 effectively becomes a target of 10, so the success rate rises to fifty-five percent.

This shows what one point of bonus is worth. A plus one always adds five percentage points, but the relative improvement grows as the base chance shrinks. At a ten percent success rate, plus one is a fifty percent improvement; at fifty percent, it is a tenth.

• Target 5 or higher: 80 percent
• Target 10 or higher: 55 percent
• Target 11 or higher: 50 percent
• Target 15 or higher: 30 percent
• Target 18 or higher: 15 percent
• Target 20: 5 percent

How strong is advantage

Advantage means rolling two D20s and keeping the higher. You only fail if both rolls fail, so the success chance is one minus the failure chance squared.

Start from a fifty percent chance and advantage lifts it to seventy-five percent. That is twenty-five percentage points, equivalent to a plus five bonus. This is exactly where advantage is worth the most.

At ninety percent, advantage only reaches ninety-nine percent, a gain of nine points. At five percent, it reaches 9.75 percent, a gain of under five points. Advantage is at its strongest on genuinely uncertain checks and much weaker on easy or near-impossible ones.

Advantage and disadvantage as averages

A single D20 averages 10.5. Take the higher of two and the average rises to about 13.83; take the lower and it falls to about 7.17.

So advantage is worth roughly plus 3.3 on average. That figure is only an average, though, and the real effect swings substantially with the target number.

That asymmetry is what makes advantage more interesting than a flat bonus. Its value changes with the situation, so players have to think about when to spend it.

Critical hits and fumbles

Many rule sets treat a natural 20 as an automatic success and a natural 1 as an automatic failure. Each is five percent, so every check keeps at least a five percent chance of going either way.

With advantage, the chance of rolling a 20 goes from five percent to 9.75 percent, close to double. Where criticals are powerful, advantage is worth more than the plain success-rate math suggests.

Disadvantage does the same in reverse, pushing the chance of a natural 1 to 9.75 percent. In systems with harsh fumble rules, disadvantage is more dangerous than it looks.

Why damage uses several smaller dice

Checks use a single D20, but damage typically uses several D6s or D8s. That is a deliberate split, because checks want dramatic swings and damage wants predictability.

Summing several dice pulls results towards the centre. Three D6s land near ten or eleven most of the time and almost never on three or eighteen, so damage becomes something you can plan around.

If damage also used a single D20, you would see ones and twenties constantly and combat would be impossible to reason about. Choosing which kind of variance you want is the foundation of dice-based rule design.