# 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**.

```dice
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))
```

## 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](https://www.dndbeyond.com/sources/dnd/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](05-roll-under.html) · [Next: one comparison table](07-probability-tables.html).
See the [ordinary ability-check recipe](../cookbook/dnd2014-ability-check.html)
for a reusable, explicitly non-attack model.