Understanding RPG Dice Probability: d20 Odds, Bell Curves and Charts

Most disagreements about difficulty at a table are really disagreements about arithmetic. A referee sets a target, a player groans, and nobody involved has a d20 probability chart in front of them. The numbers are not complicated, and knowing them changes how a scene is written as much as how it is adjudicated.

Every dice mechanic answers one question: given this roll, how often does the attempt succeed? Everything filed under rpg dice probability comes back to that one question. Answering it for a single flat die takes a subtraction. Answering it for a pool of dice takes a little more, and that is where a dice probability chart stops being a curiosity and starts being a design tool.

What rpg dice probability really describes

A distribution is not a difficulty setting. It is the shape of the outcomes a mechanic can produce and how often each one appears. Two systems can share an average and behave completely differently, which is why rpg dice probability is worth reading as a picture rather than a single number.

The practical consequence shows up in how a system feels. Flat dice produce swing: competence rarely protects anyone from an embarrassing result. Curved dice produce reliability: the expert usually performs like an expert, and the surprise, when it lands, is genuinely a surprise. Neither is better. They are different promises to the people at the table, and a referee who knows which promise the system makes can set targets that keep it.

The d20 is a flat line

A twenty-sided die has no centre. Each face turns up 5% of the time, so the chance of meeting or beating a target is simply the count of faces at or above it. Nine faces reach 12 or higher, giving 45%. Six faces reach 15, giving 30%. One face reaches 20, giving 5%.

This produces the single most useful fact in d20 play: every point of target number is worth exactly five percentage points, everywhere on the scale. Moving a target from 13 to 15 costs the same ten points whether the character is hopeless or heroic. No other common mechanic is that well behaved, and it is why d20 modifiers can be balanced by hand.

Reading a d20 probability chart

A d20 probability chart is a straight line, and straight lines are easy to keep in your head. Subtract the target from 21 to get the number of successful faces, then multiply by five. Target 8 leaves thirteen faces and 65%. Target 18 leaves three faces and 15%.

The chart only misleads in one place, and it is worth naming: modifiers do not scale. Adding +2 to a roll always shifts the odds by ten points, but if the target already sits below the modifier the bonus buys nothing, because the roll could not fail anyway. Bonuses stacked onto easy tasks are wasted arithmetic.

Where the bell curve changes everything

Sum three six-sided dice and the flat line collapses into a bell curve. Ten and eleven appear 12.5% of the time each. Three and eighteen appear 0.46% of the time — once in 216 rolls. Just under half of all results, 48.15%, land between 9 and 12.

A bell curve compresses the interesting range and makes modifiers behave unevenly. Near the middle a +1 is worth a lot, because it moves the roll through the crowded part of the distribution. Out at the edges the same +1 is nearly worthless. Systems built on a bell curve therefore use small modifiers and mean them: GURPS lives on 3d6, and a +2 there is a significant advantage rather than a rounding error.

Odds with advantage and disadvantage

Rolling twice and keeping the higher die is not a flat bonus, and treating it as one leads to bad estimates. The gain depends entirely on where the target sits. At target 11 the odds move from 50% to 75%, a swing of twenty-five points. At target 5 they move from 80% to 96%, sixteen points. At target 20 they move from 5% to 9.75%, under five points.

Disadvantage mirrors it. A 50% chance falls to 25%; a 30% chance falls to 9%. The mechanic is at its most dramatic on coin-flip rolls and almost inert on the desperate ones, which is the opposite of how players tend to describe it. If a scene is meant to turn on whether advantage was earned, the target has to sit near the middle of the range for the advantage to matter.

Ability scores are a distribution too

Character generation is dice probability wearing a different hat. Straight 3d6 down the line gives the same bell described above, with an average of 10.5 and about one score in eleven at 15 or better. Rolling four dice and discarding the lowest raises the average to 12.24 and pushes 23.1% of scores to 15 or above.

The difference is not a small houserule. It is the gap between a game about capable people and a game about ordinary ones, decided before play begins by which method the table picked. Most groups want the first of those games, which is why the common default is 4d6 drop lowest. Mean 12.24. An 18 appears in 1.62% of scores, and 10.5% land at 8 or below.

Building a dice probability chart of your own

Nothing about rpg dice probability requires heavy machinery. For flat dice, arithmetic is enough. For anything pooled, exploding or step-based, work it out properly: enumerate the outcomes, count the ones that succeed, divide. A short script handles this in seconds and, unlike intuition, it does not round in the direction you were hoping for. AnyDice is the standard tool for the job and will draw the distribution for you.

Whatever you use, keep the output as a dice probability chart rather than a single average. A designer who checks the tails catches the failure modes early — the mechanic that produces a catastrophe 8% of the time, or the one that cannot fail once a character reaches a certain rating. Those are the results that decide whether a system survives contact with a table, and an average will never show them to you.

Common questions