Call of Cthulhu success bands — the d100 has six outcomes, not two

The closed-form ratios for skills divisible by 5

For any skill S that's a multiple of 5 — the entire canonical CoC skill range — the floor in the threshold formulas is exact. So:

P(extreme) / P(regular) = (S/5) / S = 1/5
P(hard)    / P(regular) = (S/2) / S = 1/2
P(critical) / P(regular) = (1)  / S = 1/S

"Extreme is one-fifth of regular" and "hard is one-half of regular" hold across S ∈ {5, 10, 15, ..., 95} exactly. The pillar prose can reach for these as anchors instead of recomputing from each skill. The third ratio gives a sense of how rare critical success is relative to a regular pass — a S=80 character criticals on 1/80 of their successes; a S=20 character on 1/20. Lower-skill characters get critical successes proportionally more often per regular pass, even though their absolute critical rate is the same 1/100.

For non-multiples of 5 the ratios are off by at most one in the numerator. At S=51, ⌊51/5⌋ = 10, so P(extreme) = 10/100, P(regular) = 51/100, ratio ≈ 0.196. The published CoC skill ladder uses multiples of 5, so this corner case rarely matters in practice.

The fumble cliff at skill 50

The fumble rule has two cases:

• S < 50: fumble = roll ≥ S + 50. At S=10, fumble starts at 60 (41/100 outcomes). At S=49, fumble starts at 99 (just 2/100).
• S ≥ 50: fumble = roll ≥ 96, a flat 5/100 regardless of skill. At S=50, S=70, S=99 — same 5/100 fumble rate.

Cross the 50-skill threshold and the fumble probability jumps up:

It's the only place in the skill range where investing in your skill makes catastrophic failure more likely. The mechanic doesn't make it narrative-natural sense — it's a clean rule cliff, the kind of detail that's invisible in build-guide prose but matters once you start counting outcomes. A character building from S=45 to S=50 raises their regular-pass rate by 5 points (45 → 50) and their fumble rate by 4 points (1 → 5).

Below S=50, fumble is dominated by the S + 50 formula and shrinks rapidly as skill rises. Above S=50, fumble is a flat 5%. So the shape of risk changes as well as the magnitude: low-skill characters are mostly failing-to-fail (regular fail vastly outweighs catastrophic fumble), high-skill characters almost never fail-without-fumbling — at S=99, P(fail) = 0, P(fumble) = 5/100, P(regular) = 95/100. Every "fail" is a "fumble" in the limit.

How the bands compose with damage rolls

CoC damage is rolled separately from the skill check, but the band the check landed in modulates the damage. Common rule conventions:

• Regular success — roll the weapon's damage normally.
• Hard success — usually irrelevant for damage outside opposed checks; matters for narrative gating.
• Extreme success — apply damage modifier (DB) max, or roll-and-add max die for some weapons; some keepers rule "max damage" outright.
• Critical (01) — typically maximum damage and often a narrative bonus (impale, knockdown, etc.).

The damage distribution is composable with the existing engine grammar — once roll_under SKILL ships as parser surface, you'll be able to write roll_under 65; on extreme: 1d6+1d6+db as a single expression. The analyzer already exists: call Diceplots.Engine.percentile_test(skill, die_mod) directly and compose the band probabilities with whatever damage roll the keeper rules apply.

The engine surface

The analyzer is one function call returning all six band probabilities as both f64 for display and exact-rational strings for the math:

{:ok, r} = Diceplots.Engine.percentile_test(50)
r.p_regular_exact   #=> "1 / 2"
r.p_fumble_exact    #=> "1 / 20"

{:ok, r} = Diceplots.Engine.percentile_test(49)
r.p_fumble_exact    #=> "1 / 50"   (the cliff: 2/100 vs 5/100 at 50)

{:ok, r} = Diceplots.Engine.percentile_test(50, :bonus)
r.p_regular_exact   #=> "3 / 4"    (bonus die at S=50)

Skill is range-checked 0..=100. The die_mod argument defaults to :none (plain d100) and accepts :bonus or :penalty for the shared-units order-statistic die mechanics. The full bonus / penalty math lives in its own pillar.

Where this matters in practice

The 6-band model surfaces design and play decisions that the "did I succeed?" binary lens hides.

Skill investment past 50. Going from S=49 to S=50 is a +5 to the regular-pass rate, but a +4 to the fumble rate too. The character builds up "competence" and "risk of catastrophe" simultaneously. For Investigator-class skills (Library Use, Spot Hidden), the fumble cost is usually narrative — finding the wrong book or jumping at shadows. For combat skills (Firearms, Brawl), fumble usually means weapon malfunction or hitting the wrong target, which is a real mechanical cost. The "stop at S=49" tactic is a known min-maxer move; the math is in this table.

Levels-of-success scaling damage. Most CoC 7e weapons have damage rules that scale with the success band (extreme = max damage, etc.). If you're tuning encounters, the ratio P(extreme) / P(regular) = 1/5 is your "how often does the maximum damage outcome fire" anchor. Five regular hits on average contain one extreme. Designing a fight around "the PCs land an extreme on this turn" is a 1/5 conditional probability, not a raw 1/10 or 1/15 on the d100.

Cross-system: any d100 roll-under. RuneQuest, Delta Green, Pendragon, Mythras — the band thresholds shift (RuneQuest's "special success" is at S/5 like CoC's extreme; Delta Green's "skill check" reads cleaner at S/2 = hard) but the cumulative-success / exclusive-failure structure is shared. The engine's percentile_test shape will parameterise cleanly when those systems ship; the math here transfers.