What a skill check really measures

A twenty-sided die is a familiar object in tabletop role-playing: a solid with twenty faces, which this note treats as a fair die, each face equally likely to land up. Because every face is equally likely, a single roll is about as simple as probability gets, and yet the way games turn that roll into a “skill check” hides several decisions that matter more than the die does.

This note works through those decisions one at a time. The calculator at the end lets you try your own numbers without trusting mine.

The raw die is flat, and that is the point

Roll one twenty-sided die and there are twenty outcomes, one for each face, and each outcome happens with probability 1 in 20 — five per cent. The set of outcomes and their probabilities is what the mathematics calls a discrete probability distribution (OpenStax — Mean or Expected Value and Standard Deviation). For this die the distribution is flat: no face is more likely than another, so the chart of outcomes is a horizontal line.

In the free rules for the 2024 core rulebook, the d20 is the resolution engine: whenever an action’s result is in doubt, the game calls for a d20 roll (D&D Beyond — Free Rules: Playing the Game). The flat distribution is why the d20 is easy to reason about in that role. Every result from 1 to 20 is equally reachable, so a player can read “16 or higher” as “five faces out of twenty” without doing any probability work at all.

Roll under a target number

One common form of check is roll under: pick a target number, roll the die, succeed if the result is at or below it. If the target is 8, the successful faces are 1, 2, 3, 4, 5, 6, 7 and 8 — eight faces. The chance is 8/20, which is 40%.

Two edge cases quietly change that number.

• Strictly under. If the referee says “roll under 8” and means 7 or less, the chance is 7/20, or 35%. The word “under” is doing real work; a table that never settles it will argue about it eventually.
• Boundary rules on natural rolls. Some tables add a house rule that a natural 1 is a special failure and a natural 20 a special success on every check. (The free rules do not do this for ability checks; they give the natural 20 and natural 1 rule to attack rolls.) If a natural 20 always succeeds, a roll-under target of 19 leaves a 100% chance because every face succeeds. If a natural 1 always fails, even a target of 20 tops out at 95%.

Neither edge case is exotic. Both are the kind of detail that a reader of a rulebook skims and a designer cannot.

Roll over a target number

The mirror image: succeed if the die lands at or above a target. This is the form the free rules use: roll a d20, add modifiers, and the test succeeds if the total equals or exceeds the target number, called a Difficulty Class for ability checks (D&D Beyond — Free Rules: Playing the Game). A target of 13 succeeds on 13 through 20, which is eight faces out of twenty, or 40%, the same 40% as rolling 8 or lower, because the twenty faces can be mirrored around the midpoint. Once you notice the mirror, half of a rulebook’s modifiers become easier to read: raising a “roll over” target by one is exactly the same difficulty change as lowering a “roll under” target by one.

The modifier trap

Most games let a character’s training shift the target rather than the die. A +4 bonus on a roll-under target of 8 raises it to 12, and the chance goes from 40% to 60%.

That step is twenty points for four points of modifier. It feels generous until you look at the edges. At a target of 18 (90%), a +4 modifier would push the target to 22, but the die cannot show 22; the real cap is 20, or 100%. Four points of modifier that were worth twenty percentage points in the middle of the range are worth ten percentage points here, because only two faces were left to gain. At a target of 20 or more a modifier adds nothing at all.

The arithmetic is simple, but the effect is not: the same modifier is worth between zero and twenty percentage points depending on where the character already sits. A designer who says “a +4 bonus for expertise” is really saying something different for a beginner and for an expert.

Advantage in one die, and its arithmetic

The free rules define Advantage as rolling two d20s and keeping the higher (D&D Beyond — Free Rules: Rules Glossary). Against a roll-over target, a roll with Advantage succeeds if either die does; in this note’s roll-under terms the matching rule keeps the lower die, which again succeeds if either die does. For a roll-under target of 8, the chance of failing once is 12/20, or 0.6. Failing twice in a row is 0.6 × 0.6 = 0.36. So the chance of succeeding at least once is 1 − 0.36 = 0.64, or 64%.

Convert that back to a modifier and the inner trap appears: 64% is closest to a target of 13 (65%) on the plain d20. Advantage on this target is worth roughly five points of modifier (40% to 64% is 24 points, or about 4.8 faces). In the middle of the range, that is more than a +2 or +3 bonus would give.

The “equivalent flat target” column is the target that gives a similar chance on an unmodified die. It is approximate on purpose; the table exists to show the direction of the effect, not to replace a calculator.

Where the arithmetic stops helping

Nothing above says a d20 is the right die. A flat distribution makes every step of the range equally likely, which produces a particular feel: wild swings, lucky rookies, fumbles at the worst moment. A sum of several dice clusters toward the middle and thins the tails, which makes outcomes steadier; spread is the subject of a separate note.

A d20 check measures what the table has decided to measure: which numbers count, whether ties or natural rolls change the answer, and where a bonus is applied. The die itself contributes one thing only: twenty equally likely faces.

Look up the numbers

To read a roll-under check, add the modifier to the target, cap the result at 20 and floor it at 0, then look up the adjusted target below. Advantage is not included; square the failure chance as shown above.

For example, a target of 8 with a +4 modifier is an adjusted target of 12, which succeeds on 12 of 20 faces, or 60%. A target of 18 with the same modifier would be 22, which is capped at 20, or 100%.

Working notes

The probability of a single face on a fair die with n faces is 1/n; for a twenty-sided die that is 0.05. Any “roll under” or “roll over” target can be read by counting faces, and that count divided by 20 is the chance. Advantage multiplies the failure probability by itself. Modifiers move the target, and a target outside 1–20 collapses to a certainty in one direction. The spread of a distribution, which becomes the interesting question once dice are summed, is measured by standard deviation (OpenStax — Mean or Expected Value and Standard Deviation).

Method

Every percentage is a face count divided by 20. Advantage chances are 1 − (failure chance)², and the “equivalent flat target” column is the plain d20 target whose chance is closest to the advantage chance (for example 64% against 65% for a target of 13). Counts were checked by script. The description of the d20 test, Difficulty Class and Advantage was read on 2026-10-31 on the pages listed below.

Sources

• D&D Beyond — Free Rules: Playing the Game: the d20 Test, target numbers and Difficulty Class, and the natural 20 and 1 rule for attack rolls.
• D&D Beyond — Free Rules: Rules Glossary: the definition of Advantage.
• OpenStax — Mean or Expected Value and Standard Deviation: outcomes and their probabilities as a distribution, and standard deviation as the measure of spread.

Written for: a player who is about to design a character and wants to know what the percentages on a character sheet are really promising.