# Round each result, not the average

**Part E · L20 · Prerequisite: L19 · One modelling idea**

## Situation, question, and prediction

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.

## Build

`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)
```

## Run, read, and check

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.

## Change one thing

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

## Try it yourself

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.

## Rules and model notes

[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.

## What you now know / where next

Round each result, not the average is the reusable idea. Follow the generated
previous/next links below, or return to the [course index](index.html).
For a complete self-contained application, see [fireball-half-damage](../cookbook/fireball-half-damage.html).