Open this document in the playground · Download source
An average is not a success chance
Part B · L06 · 5–10 minutes · Prerequisite: L05
The situation and question
In D&D 2014, the DM calls for a Strength (Athletics) check to climb a difficult surface. The Difficulty Class (DC) is 15. Two characters have total check bonuses +2 and +5, already including any applicable proficiency. Which has the better chance on this single attempt, and how much better is it?
For an ordinary ability check, roll a d20, add the bonus, and meet or beat the DC. These are not attacks: do not add automatic failure on a natural 1 or automatic success on a natural 20. This example excludes advantage, spells, rerolls, passive checks, and special class features.
Predict
The +5 character's average total is above 15. Does that mean they usually succeed, always succeed, or need no roll? The DM still decides whether a check is called for; our calculation only answers the stated uncertain attempt.
Build
We reuse assignment, addition, and .p_ge from Part A. Each roll variable is a
distribution of totals. Report the totals to see their ranges and means, then
ask a separate question about success. Four outputs are useful here because
we are deliberately contrasting two measurements for two choices.
dc = 15
bonus_a = 2
bonus_b = 5
check_a = d(20) + bonus_a
check_b = d(20) + bonus_b
output("Character A: check totals", check_a)
output("Character B: check totals", check_b)
output("Character A: chance to meet DC", check_a.p_ge(dc))
output("Character B: chance to meet DC", check_b.p_ge(dc))
Character A: check totals · DieRoll · mean 12.500
| outcome | % | frac | X/20 |
|---|---|---|---|
| 3 | 5.00 | 1/20 | 1 |
| 4 | 5.00 | 1/20 | 1 |
| 5 | 5.00 | 1/20 | 1 |
| 6 | 5.00 | 1/20 | 1 |
| 7 | 5.00 | 1/20 | 1 |
| 8 | 5.00 | 1/20 | 1 |
| 9 | 5.00 | 1/20 | 1 |
| 10 | 5.00 | 1/20 | 1 |
| 11 | 5.00 | 1/20 | 1 |
| 12 | 5.00 | 1/20 | 1 |
| 13 | 5.00 | 1/20 | 1 |
| 14 | 5.00 | 1/20 | 1 |
| 15 | 5.00 | 1/20 | 1 |
| 16 | 5.00 | 1/20 | 1 |
| 17 | 5.00 | 1/20 | 1 |
| 18 | 5.00 | 1/20 | 1 |
| 19 | 5.00 | 1/20 | 1 |
| 20 | 5.00 | 1/20 | 1 |
| 21 | 5.00 | 1/20 | 1 |
| 22 | 5.00 | 1/20 | 1 |
Character B: check totals · DieRoll · mean 15.500
| outcome | % | frac | X/20 |
|---|---|---|---|
| 6 | 5.00 | 1/20 | 1 |
| 7 | 5.00 | 1/20 | 1 |
| 8 | 5.00 | 1/20 | 1 |
| 9 | 5.00 | 1/20 | 1 |
| 10 | 5.00 | 1/20 | 1 |
| 11 | 5.00 | 1/20 | 1 |
| 12 | 5.00 | 1/20 | 1 |
| 13 | 5.00 | 1/20 | 1 |
| 14 | 5.00 | 1/20 | 1 |
| 15 | 5.00 | 1/20 | 1 |
| 16 | 5.00 | 1/20 | 1 |
| 17 | 5.00 | 1/20 | 1 |
| 18 | 5.00 | 1/20 | 1 |
| 19 | 5.00 | 1/20 | 1 |
| 20 | 5.00 | 1/20 | 1 |
| 21 | 5.00 | 1/20 | 1 |
| 22 | 5.00 | 1/20 | 1 |
| 23 | 5.00 | 1/20 | 1 |
| 24 | 5.00 | 1/20 | 1 |
| 25 | 5.00 | 1/20 | 1 |
Character A: chance to meet DC · Prob
| outcome | % | frac | X/20 |
|---|---|---|---|
| Character A: chance to meet DC | 40.0 | 2/5 | 8 |
Character B: chance to meet DC · Prob
| outcome | % | frac | X/20 |
|---|---|---|---|
| Character B: chance to meet DC | 55.0 | 11/20 | 11 |
Run and read
| Character | Range | Mean total | Successful natural faces at DC 15 | Chance |
|---|---|---|---|---|
| A, +2 | 3–22 | 12.5 | 13–20: eight faces | 40% |
| B, +5 | 6–25 | 15.5 | 10–20: eleven faces | 55% |
The +5 choice improves this chance by 15 percentage points, not “three percent” because its mean is three higher. A mean is measured in points on the roll; a probability is a proportion of attempts. Character B can still fail 45% of the time despite a mean above the DC.
For these identical dice with different flat bonuses, the larger bonus cannot reduce success at a fixed DC. That does not make a mean a universal substitute for a threshold query when other distributions or rules change.
Change one thing
Raise dc to 20. Check: A succeeds on 18–20 (15%); B on 15–20 (30%). The
means do not change: the rule for rolling is unchanged; only the question changed.
Try it yourself
For bonus +5, compare DC 25 and DC 26. Answer check: only natural 20 meets DC 25, giving 5%; no face meets DC 26, giving 0%. A natural 20 is not an automatic success for this ordinary ability check. At DC 6 every face succeeds, including natural 1. Do not transfer those boundary conclusions to an attack roll.
What you now know
Pick the measurement that answers the decision. Mean total and success chance are useful, different quantities; neither describes every possible outcome.
Rules and model notes
Source: D&D Basic Rules 2014, Using Ability Scores, Ability Checks, Skills, and Strength Checks, checked 2026-09-15. The total integer bonus is an input, not a model of character creation. This report does not decide eligibility, repeated attempts, or the cost of failure.
Where next
Previous: roll-under · Next: one comparison table. See the ordinary ability-check recipe for a reusable, explicitly non-attack model.
Content ID: L06 · Review status: pilot
Prerequisites: L05
Rules review: source checked 2026-09-15; human review pending