Expected strikes to kill — when mean wins back from variance

The setup

Until now we've asked "what's the chance one strike finishes the target?" — call it p₁ = P(damage ≥ HP). The natural next question: how many strikes does it take, in expectation? And what's the chance of finishing in ≤ N?

Two reasons it matters:

• Long fights. Bosses rarely die in one swing. The expected-strike count is the number you actually plan around.
• Action economy. "How many rounds until this dies" decides whether to chip or to nuke, whether to use a one-shot consumable, whether to fish for a crit or just keep swinging.

The math, briefly

Each strike is i.i.d., so total damage after N strikes is the N-fold convolution of the per-strike distribution with itself. Let p_n = P(damage in n strikes ≥ HP). Then:

P(killed in exactly n strikes) = p_n − p_n−1
E[N] = ∑_n≥1 P(N ≥ n) = 1 + ∑_n≥1 (1 − p_n)

The second line is the survival-function identity for a positive-integer random variable: expected value equals the sum of survival probabilities. It converges fast — once p_n approaches 1, the tail contributes almost nothing.

Worked example: 2d6+5 across an HP sweep

Greatsword swing, +5 strength modifier. Single strike has mean 12, max 17, variance ≈ 5.83. Hand-derived from the engine's exact rationals:

Notice: as HP doubles past the single-strike max, E[N] roughly doubles too. The slope is 1/mean = 1/12. That's the renewal-theorem rule.

The renewal-theorem rule of thumb

For HP much larger than the single-strike maximum, the elementary renewal theorem gives:

E[N] ≈ HP / μ + (σ² / 2μ² + 1/2)

where μ is the per-strike mean and σ² the per-strike variance. The first term — HP/μ — dominates and grows linearly. The second term is a bounded correction: variance contributes a small offset, but doesn't compound with HP. For long fights, mean is king.

For 2d6+5 (μ=12, σ²≈5.83) at HP=240: rule of thumb predicts 240/12 + (5.83/288 + 0.5) ≈ 20 + 0.52 = 20.52. Engine says 20.479. Inside 0.05 strikes — for a quantity that took 35 N-fold convolutions to compute exactly, that's a pretty good back-of-envelope.

Variance washes out: reliable vs nuke at scale

The reliable-vs-nuke pillar paired 3d4+4 (reliable, mean 11.5, variance 3.75) with 1d12+5 (nuke, mean 11.5, variance 11.92). Same mean, triple the variance. Single-strike kill chance flipped depending on HP: reliable wins below the mean, nuke wins above. Now stretch HP out and watch the gap close:

At HP 120, the strikes-to-kill chains for these two completely different distributions are within 0.3% of each other. The variance that mattered so much at HP 15 has been absorbed into the constant term of the renewal series. Side-by-side at HP 120:

The two regimes — when each pillar's lesson applies

The variance pillar and this one aren't contradicting each other; they describe two different scaling regimes:

• HP ≲ single-strike maximum — variance-pillar territory. The kill turns on the right-tail mass. Higher variance helps you when the target's HP sits just out of mean's reach; hurts you when the target is well within reach.
• HP >> single-strike maximum — this pillar's territory. Variance contributes a bounded correction (the σ²/2μ² + 1/2 offset), and otherwise mean wins linearly. Doubling your mean halves your expected strikes; halving your variance shaves a fraction of one strike.

The boundary is roughly HP ≈ 1.5 × single-strike max, give or take. Below: think variance. Above: think DPR.

Where this matters in practice

• Boss fights. A 200 HP boss vs a level-5 fighter swinging 2d6+5 (mean 12) is firmly in the renewal regime — expected ≈ 16-17 strikes. Build choices that buff mean (Great Weapon Master's +10 damage on demand, Smite, magic weapons) directly cut that count. Build choices that buff variance (Champion crit range, Reckless advantage as variance) contribute a much smaller correction.
• Action economy and Extra Attack. Going from one attack per round to two doesn't just double DPR — it halves per-round variance via the multi-roll average, pulling you toward the renewal regime faster. The combat resolves in fewer rounds AND with less round-to-round swing.
• Diablo / PoE / Last Epoch boss-burning. Mapping builds (lots of small enemies) live in the variance regime — crit-stacking, fat-tail builds clear faster. Boss-rush builds should optimize for sustained DPR — the boss's HP bar is the renewal regime made literal.
• Resource economy across many fights. Knowing E[N] for typical encounters tells you, in expectation, how many spell slots / bonus actions / consumables a long day will burn. DM-side: easier than per-fight simulation.

Try it yourself

↦ /strike/2d6+5?hp=60 — slide the HP, watch E[N] climb linearly ↦ /vs/3d4+4/1d12+5?hp=120 — same mean, watch the chains converge ↦ /vs/2d6+5/1d12+5?hp=120 — different means, gap stays open