Why 2d6+5 beats 3d4+4 against an 11 HP target — even with nearly identical means

The setup

Two damage rolls. Same exact world: an enemy at 11 HP, one swing left. Pick one:

• 2d6+5 — range 7–17, mean 12, variance ≈ 5.83
• 3d4+4 — range 7–16, mean 11.5, variance ≈ 3.75

The means differ by half a point. The minimums are identical. The first roll caps slightly higher. Most players would describe these as basically interchangeable.

They're not.

The number that matters

At 11 HP, the relevant statistic isn't the mean — it's the probability that the roll lands at or above 11. Worked out from the exact rationals:

• 2d6+5 finishes the target in 13/18 ≈ 72.22% of swings. There are 36 equally-likely outcomes for 2d6; 26 of them add to 6 or more (the threshold for clearing 11 with a +5 modifier).
• 3d4+4 finishes the target in 11/16 = 68.75% of swings. There are 64 equally-likely outcomes for 3d4; 44 of them add to 7 or more.

That's a 3.5-percentage-point gap, and it goes the opposite direction from what the means suggest. The lower-mean roll wins.

Why

Both means hover just above the threshold (12 and 11.5 vs HP 11). When the mean is close to the threshold, the kill-probability question reduces to one thing: given that the typical roll is around the threshold, what's the probability of landing on the high side?

For two distributions with similar means, the one with greater variance spreads more probability mass both below and above the mean. But only the "above" side counts toward a kill. Spreading 50% of the mass higher is a strict win when 50% above is what you need.

The intuition flips the moment your mean sits comfortably above HP. If you're swinging for 25 average against an 11 HP target, you're going to one-shot it 99% of the time regardless of variance. In that regime you'd actually prefer the lower-variance roll, because there's a remote chance the high-variance one rolls catastrophically low.

The rule: kill probability is non-decreasing in variance when mean ≤ threshold, and non-increasing when mean ≥ threshold. The crossover sits at the mean itself.

Where this matters in practice

Tabletop D&D is the default audience and the cleanest example, but the math is rule-system-agnostic. Same conclusion applies to every dice-driven combat system that asks "do I drop this target this round."

Baldur's Gate 3 weapon picks at low levels. When you're chip-damaging an enemy whose HP sits just above your average per-swing damage, the higher-variance weapon outperforms. The trade reverses past mid-game once your modifiers push the mean comfortably over the enemy HP — at that point you're one-shotting most things anyway and the lower-variance weapon eats fewer bad rolls.

Diablo 4 / Path of Exile 2 crit-fishing. High-crit-chance builds are higher-variance per swing. That helps when individual swings aren't already overkill, and hurts when they are. Same crossover at the swing-mean = enemy-HP point.

Reliable vs nuke debates more generally. Multi-hit cleave (low variance, similar mean) loses to single-big-swing (high variance, similar mean) in the breakpoint-bound regime where exactly one swing has to threshold the kill.

Auto-hit vs attack-roll spell choice. Magic Missile vs Scorching Ray is the same lesson applied to a slot-cost decision. MM's tight 3d4+3 distribution wins kill probability below the spell's mean; Scorching Ray's binomial-of-rays high variance wins above.

Where the lesson stops applying. Against boss-grade HP (well past your single-strike maximum), variance washes out into a bounded constant. See finishing Strahd for that regime, where mean is back in charge.

Try it yourself

↦ /vs/2d6+5/3d4+4 — interactive comparison ↦ /strike/2d6+5 — single-strike full distribution ↦ /kill/2d6+5/11 — single-question P(finish)

Or change the HP in any of those URLs (or the input on the page) — drop it to 8 and the math swaps. Both rolls finish the target almost always, and now the lower-variance 3d4+4 has the slight edge (one-shot reliability beats high-roll potential).