Save-for-half — when does Fireball's floor(damage / 2) actually decide the fight?

The mechanic, briefly

Per RAW (PHB Saving Throws section): when a damage spell calls for a saving throw, roll the damage dice once. Each target makes their save. On a failed save, the target takes the full rolled damage. On a successful save, the target takes floor(damage / 2) — half of the same total, not half the dice rolled separately.

Subtlety: the floor is on the total, not per-die. A Fireball that rolls [1, 6, 3, 5, 4, 2, 6, 1] = 28 deals 28 on fail and 14 on save, not "round each die down then sum." The engine matches RAW: the per-attempt distribution mixes the full dmg distribution with a per-outcome floor(x/2) transformation of the same.

Save probability is the standard 5e formula: P(save) = clamp((21 + save_bonus − DC) / 20, 0, 1). No nat-1 auto-fail, no nat-20 auto-success on standard saves (that's for attacks; death saves are separate).

The headline panel — Fireball at DC 15 vs +5 DEX

Standard 3rd-level slot Fireball (8d6) against a CR-5 monster with +5 DEX save (P(save) = 11/20 = 55%). The left panel shows the raw 8d6 distribution (mean 28, range 8–48); the right shows the post-save distribution (full mass on fail, halved mass on save):

The post-save distribution has support down to 4 (floor(8/2) = the all-1s save outcome) — half-on-save means the lower half of the support fills with 55% of the mass weight. Mean drops from 28 (always full) to roughly 20.3 (55% half + 45% full = 0.55·14 + 0.45·28 ≈ 7.7 + 12.6 ≈ 20.3, modulo floor effects).

The DC table — where the spell lands

P(save) varies linearly with the spread between save bonus and DC. Mean per-target damage is P(fail)·E[dmg] + P(save)·E[floor(dmg/2)] — for symmetric dice the expected halved-damage approximates E[dmg]/2 with a small floor correction.

Pattern: every 5 points of DC over the target's bonus halves the save chance, which roughly halves the save-erasure of damage. A Fireball at DC 15 vs +0 (low-tier minions) loses ~15% of mean damage to the save mass; vs +5 (CR-5 monsters) it loses ~28%. Push the DC, recover most of the loss.

Save-or-suck — the binary cousin

Some spells use the same save mechanic but with a binary outcome: full effect on fail, nothing on success. Hold Person, Banishment, Tasha's Hideous Laughter, lots of cantrip-tier controls. The engine models this as the zero on-success flag. P(save) becomes the probability the spell does nothing; P(fail) is the probability of full effect.

Per-cast expected damage for save-or-suck is just P(fail) · E[dmg] — no half-mass redistribution, just a binary on/off. Compare:

The save-or-suck panel has 9/20 = 45% mass at 0 (the save succeeded) and the remaining 55% spread per the 1d6 distribution. No half-zone in between.

Trade-off: save-or-suck has higher per-failed-save payoff (full effect with no half-tax) but a hard zero on success. Save-or-suck dominates when P(save) is low — against poor-save targets (CR-1 humanoids, swarms), the binary nature of the effect eats a smaller share of the expected value than save-for-half does.

Upcasting — the slot tradeoff

Fireball adds 1d6 per slot level above 3rd. Upcast at 5th (10d6), at 7th (12d6), etc. Each die adds mean +3.5 to raw damage — but the save-for-half tax reduces the effective contribution. At DC 15 vs +5 save (P(save) = 55%), each added die contributes 0.45·3.5 + 0.55·1.75 ≈ 2.54 effective mean per cast.

Upcasting Fireball from 3rd (8d6) to 5th-level slot (10d6) ALSO usually means a higher caster level → typically +1 to DC (for prepared casters who upcast at higher proficiency). The DC bump compounds: more raw damage AND a smaller P(save). The right panel shows 10d6 at DC 17 — both effects stacked.

The slot-level question players ask: "Should I upcast Fireball to 5th level instead of casting Cone of Cold (8d8 at DC same)?" Engine answer: at DC 17 vs +5 save, 10d6 (mean ~28) vs 8d8 (mean ~31.5) gives Cone the raw edge — but Cone's tighter single-die-size distribution has lower variance, which matters less per-cast than total damage. Both are within ~10% mean DPS per slot.

The AOE multiplier — why DEX +0 minions are Fireball food

Per-cast damage scales with target count. Per RAW the dice are rolled once and each target saves independently — so the per-cast distribution isn't a simple convolution; it's the joint distribution of one shared damage roll and N independent Bernoulli saves. The engine computes this exactly via the t<N> postfix on save expressions.

Same Fireball, three pod sizes, all at DC 15 vs +5 DEX save:

Single target (left) caps at 48; four targets (right) extends to 192. Per-cast mean grows roughly linearly with N — but the shape changes too: more targets means narrower relative spread (CLT in action across the binomial-of-saves).

Same Fireball against weak minions (DEX +0) shows the AOE payoff at its peak — P(save) drops to 30%, mean per target jumps from ~20 to ~24, and 8 of them eats most of the pod's HP in one cast:

Common scenarios, computed by the engine:

• One CR-5 monster (DEX +5): per-cast mean ≈ 20. Equivalent slot single-target spell often wins.
• Pod of 4 humanoids (DEX +1, DC 15 → P(save) = 35%): per-cast mean ≈ 92. Single-target spells lose by roughly 4×.
• Swarm of 8 minions (DEX +0, DC 15 → P(save) = 30%): per-cast mean ≈ 190. The cleric should let the wizard spend a 3rd-slot here even if MM seems "more reliable."

Rule of thumb: Fireball stops being worth the slot against ≤2 high-save targets and starts being worth it against ≥3 low-save targets. The crossover is roughly at 2-3 targets when DC vs save_bonus is even.

Try it yourself

↦ /strike/8d6 save +5 DC15 — Fireball vs single CR-5 monster ↦ /strike/8d6 save +5 DC15 t4 — Fireball into a pod of 4 ↦ /strike/8d6 save +0 DC15 t8 — Fireball into a swarm of 8 weak minions ↦ /strike/8d6 save +0 DC15 — Fireball vs DEX +0 humanoid (single) ↦ /strike/10d6 save +5 DC17 — upcast 5th-level slot ↦ /strike/8d6 save +5 DC17 — DC bump alone (Heightened Spell metamagic) ↦ /strike/1d6 save +1 DC13 zero — save-or-suck distribution ↦ /vs single vs t4 — see how the pod-size payoff unfolds

Drag the HP slider in any /vs view. The save-modulated panel's kill-probability column reflects the post-save distribution — so "P(finish 30 HP)" tells you the per-cast kill probability against a target at that HP, accounting for the save-for-half mass redistribution.