# D&D 2014 — Fireball damage given the save result

**R04 · Intermediate · Scoped component**

## Question and scope

D&D 2014 Fireball at its base spell level deals 8d6 damage, halved on a successful Dexterity save. **Given that the save succeeded**, what is mean damage? Predict whether it is exactly half of 28.

Source sections are linked under Rules below; primary-source review is recorded in the release dossier. Independent human and browser/usability review are separate gates.

## Inputs and model limits

Base spell level, 8d6; one target. Reports are conditional on success/failure, not weighted by a target’s save chance.

## Complete runnable model

`dice_pool(8, 6).sum()` efficiently builds the ordinary total. `// 2` divides each possible integer result and rounds down. This is a transformation of the entire distribution, not division of a summary number.

```dice
full = dice_pool(8, 6).sum()
half = full // 2
output("Damage given failed save", full)
output("Damage given successful save", half)
```

## Reading the report and independent checks

Full mean is 28; successful-save mean is **13.75**, not 14. Half the possible probability mass is on odd totals, where rounding loses half a point, giving a mean loss of 1/4. Half-damage range is 4–24.

## Safe changes

Use one d6 instead: full mean 3.5, rounded-half mean 1.5. Verify by writing transformed faces 0,1,1,2,2,3.

## Composition and boundary checks

Why is this not the overall expected damage to a target? Answer: it conditions on the save result. You must also know the target’s save chance; L32 builds that full scoped question. Checkpoint: keep the conditioning statement in every output label.

Do not combine separate marginal reports and assume they preserve the same
rolled faces. When an event depends on several properties of one roll, keep
those properties together inside a single classifier or pool callback.

## Rules

[D&D Basic Rules 2014, Fireball](https://www.dndbeyond.com/sources/dnd/basic-rules-2014/spells) and the basic round-down convention. Base third-level spell, one target; no resistance, vulnerability, Evasion, or other damage reductions.

## Related learning

[Lesson L20](../tutorial/20-round-each-result.html) explains the modelling idea step by step.
[Cookbook index](index.html) offers both game and mechanic views of these same recipes.