Why isn't bonus die just kh1 on a d100?

Because only the tens digit is rolled twice. CoC builds its d100 from a separate tens-d10 (00, 10, 20, ..., 90) and a units-d10 (0..9), with the convention that 00-0 reads as 100. Bonus / penalty die rolls an extra TENS-d10 only — units is shared with the original. So the order statistic operates on half the dice, and the resulting d100 distribution is not the order statistic of two independent d100 rolls. Math-wise the difference is large: full-d100 advantage at skill 50 would lift P(regular) to 75% on its own; CoC's half-advantage variant happens to land at the same 75% (a coincidence at S=50), but at skill 25 the full version would yield ~44% and CoC's variant lands at 43.5%. The two diverge more at extreme skills where the units distribution carries more of the band-membership decision.

Why does the bonus die's effect peak at skill 50?

Because skill 50 is where the band threshold sits at the modal density of the tens distribution — every tens digit (0..9) is roughly half-pass / half-fail when units randomises in. Move the threshold up or down and the tens distribution becomes already-mostly-pass or already-mostly-fail; rolling a second tens at that point doesn't help much. At S=50 the swing in P(regular) is 25 points (50 → 75 with bonus); at S=10 or S=90 it's only 9 points. The headline 'bonus die is worth +25%' is true at S=50; at the extremes you're getting +9% — substantial but a third the size.

Does the same curve hold for penalty die?

Yes, by symmetry. Penalty die at S=50 drops P(regular) from 50/100 to 25/100 — the same 25-point swing in the other direction. At S=10, penalty die drops P(regular) from 10/100 to 1/100 (9 points); at S=90, from 90/100 to 81/100 (also 9 points). The mechanic is symmetric around S=50 because the canonical d100 distribution is symmetric around the cutoff at the midpoint.

What about the critical (01) and fumble (96–100) bands?

Bonus die makes critical (roll = 01) MORE likely than 1/100, because either of the two tens rolls landing on 0 with units = 1 yields result = 1. P(critical | bonus) = P(U=1 ∧ (T1=0 ∨ T2=0)) = (1/10) · (19/100) = 19/1000. Penalty die makes critical LESS likely — only when BOTH tens are 0 with units=1, which is 1/1000 (1/10 the plain rate). Fumble at high skill (S ≥ 50, fumble = roll = 100 only): bonus die drops fumble from 1/100 to 1/1000 (10× rarer); penalty die lifts fumble from 1/100 to 19/1000 (≈1.9%, the same asymmetry as the critical case but in the opposite tail). The symmetry that holds cleanly for P(regular) doesn't extend to the band edges, where the cutoff is far from the median.

Concepts → CoC bonus / penalty die

Published 5 May 2026 · Updated 18 July 2026

Call of Cthulhu bonus / penalty die — the shared-units order statistic

The bonus / penalty die mechanic isn't kh1 / kl1 on a d100. Only the tens digit gets the order statistic; the units stays shared. Half the underlying d100 distribution is order-statistic'd; the other half is not — and the half that doesn't get the treatment carries non-trivial weight on band membership. The advantage curve peaks at skill 50 (a 25-point swing in P(regular)) and tapers to 9 points at skills 10 or 90. Companion to the success-bands pillar.

The short answer

A standard CoC d100 is rolled as tens-d10 + units-d10: T ∈ {0..9} and U ∈ {0..9}, with the convention that tens=0, units=0 reads as 100. So the d100 is uniform on 1..=100 by enumerating the 100 (T, U) outcomes.

Bonus die rolls an extra tens-d10 with units shared. Take the better (lower) of the two resulting d100 readings. Penalty die: take the worse (higher). That's not the same as full-d100 advantage / disadvantage — only T gets the order statistic, U stays the same value across both candidate readings.

Engine surface (one call, returns all six band probabilities with exact rationals):

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

{:ok, r} = Diceplots.Engine.percentile_test(50, :penalty)
r.p_regular_exact   #=> "1 / 4"

The advantage curve

Across a range of skills, the bonus and penalty die effects on P(regular) — exact rationals computed by direct enumeration of the 1000 outcomes (T1, T2, U) ∈ {0..9}³:

Skill	P(regular) plain	P(regular) bonus	+Δ vs. plain	P(regular) penalty	−Δ vs. plain
`10`	`10/100`	`19/100`	`+9/100`	`1/100`	`−9/100`
`25`	`25/100`	`87/200` (43.5%)	`+37/200`	`13/200` (6.5%)	`−37/200`
`50`	`1/2`	`3/4`	`+1/4`	`1/4`	`−1/4`
`75`	`3/4`	`187/200` (93.5%)	`+37/200`	`113/200` (56.5%)	`−37/200`
`90`	`90/100`	`99/100`	`+9/100`	`81/100`	`−9/100`

The swing peaks at S=50 with a clean 25-point Δ in either direction, drops to 18.5 points at S=25 / S=75, and flattens to 9 points at S=10 / S=90. Symmetric around the midpoint — bonus die at S=10 and penalty die at S=90 produce equivalent-magnitude shifts in the failing-mostly direction.

The intuition: bonus / penalty die's leverage comes from modulating the tens distribution. At S=50, the regular-pass cutoff sits in the middle of the tens distribution — every tens digit is roughly half-passing. Rolling a second tens and taking the better one swings the tens distribution sharply. At S=10, the cutoff is below all tens values except 0 — bonus die just helps pick T=0 more often, but units still has to land on 1..9 to pass; the leverage is smaller. Mirror at S=90.

Why "min over results" instead of "min over tens"

The literal CoC rule is "take the lower result", not "take the lower tens die". These are different when units = 0 and one of the tens is also 0 — because (T=0, U=0) reads as 100, the worst possible roll.

T1=0, T2=5, U=0: Reading 1 is 100; reading 2 is 50. Bonus die takes 50 (the lower result). Literal "lower tens" would take T=0 → 100 — the worst possible outcome, which is obviously wrong.
T1=0, T2=5, U=3: Reading 1 is 3; reading 2 is 53. Bonus takes 3. "Lower tens" agrees here.

The discrepancy lives in exactly one outcome — the (T1=0, T2=0, U=0) triple, 1/1000 of the sample space. The engine implements the canonical "min over results" semantics matching CoC 7e RAW. The headline band probabilities are unaffected by which interpretation you pick (the difference moves a 1/1000 outcome between the critical and fumble buckets), but the implementation needs to be careful enough to handle the 100-sentinel correctly.

What happens at the band edges

The clean +25/−25 symmetry that holds for P(regular) at S=50 doesn't extend to the critical and fumble bands — the cutoffs there sit at the high or low tail rather than the midpoint, and bonus / penalty asymmetry kicks in.

Band (at S=50)	Plain	Bonus	Penalty
P(critical) — roll = 01	`1/100`	`19/1000` (1.9%)	`1/1000` (0.1%)
P(extreme) — roll ≤ 10	`10/100`	`19/100`	`1/100`
P(regular) — roll ≤ 50	`50/100`	`75/100`	`25/100`
P(fumble) — roll = 100 (S ≥ 50)	`1/100`	`1/1000` (0.1%)	`19/1000` (1.9%)

At trained skill (S ≥ 50), fumble is just the natural-100 outcome — 1 specific roll of the 100. Bonus die makes critical (roll=1) 1.9× more likely AND fumble (roll=100) 10× less likely; both extremes shift in your favour symmetrically because both bands are single-outcome cutoffs at the tails. Penalty die is the mirror: critical drops 10×, fumble rises 1.9×. The 1.9 / 10 asymmetry is structural — bonus picks the lower tens (highly likely to dodge a 100 reading; harder to hit a 1 reading); penalty picks the higher tens (vice versa).

Stacking and choosing

CoC 7e bonus / penalty dice can stack (multiple bonus dice = take the lowest of N+1 tens; multiple penalties = highest); bonus and penalty cancel one-for-one before either is rolled. As a quick rule of thumb the second bonus die buys you about half the swing of the first — at S=50, P(regular) goes from 1/2 → 3/4 with one bonus die and approximately 3/4 → 7/8 with two (the order-statistic effect on the tens distribution diminishes as you stack).

Where this matters in practice

Bonus and penalty dice are the most-applied modifier in CoC play, and the swing depends sharply on where your skill sits.

Modifier value to specialists vs. generalists. A specialist with S=80 in their core skill gets relatively little from a bonus die — only +9 points to a P(regular) that's already 80%. A generalist with S=50 in everything gets +25 points where they need it most. So the "spend a luck point to get a bonus die" decision is more attractive to mid-skill characters than high-skill ones. Skill investment past 50 reduces the marginal return on bonus dice.

Penalty die is the equaliser. A penalty die at S=50 drops P(regular) by 25 points; at S=10 or S=90, only by 9 points. The same curve in reverse. Unfavourable circumstances bite mid-skill characters hardest because their skill sits at the steepest part of the bonus / penalty leverage curve. Keepers tuning a difficult atmospheric scene should keep this in mind: "take a penalty die" is closer to "your skill is reduced by 25 points" for mid-skill PCs, but only "reduced by 9 points" for the specialist who's already at S=80+.

The "spend a luck for a bonus die" budget. Players sharing one bonus die between members should give it to whoever is closest to S=50 on the relevant skill. The marginal success rate is highest there. The success-bands pillar shows the underlying skill-vs-band rates this composes with.

Common questions

Why isn't bonus die just kh1 on a d100?: Because only the tens digit is rolled twice. CoC builds its d100 from a separate tens-d10 (00, 10, 20, ..., 90) and a units-d10 (0..9), with the convention that 00-0 reads as 100. Bonus / penalty die rolls an extra TENS-d10 only — units is shared with the original. So the order statistic operates on half the dice, and the resulting d100 distribution is not the order statistic of two independent d100 rolls. Math-wise the difference is large: full-d100 advantage at skill 50 would lift P(regular) to 75% on its own; CoC's half-advantage variant happens to land at the same 75% (a coincidence at S=50), but at skill 25 the full version would yield ~44% and CoC's variant lands at 43.5%. The two diverge more at extreme skills where the units distribution carries more of the band-membership decision.
Why does the bonus die's effect peak at skill 50?: Because skill 50 is where the band threshold sits at the modal density of the tens distribution — every tens digit (0..9) is roughly half-pass / half-fail when units randomises in. Move the threshold up or down and the tens distribution becomes already-mostly-pass or already-mostly-fail; rolling a second tens at that point doesn't help much. At S=50 the swing in P(regular) is 25 points (50 → 75 with bonus); at S=10 or S=90 it's only 9 points. The headline 'bonus die is worth +25%' is true at S=50; at the extremes you're getting +9% — substantial but a third the size.
Does the same curve hold for penalty die?: Yes, by symmetry. Penalty die at S=50 drops P(regular) from 50/100 to 25/100 — the same 25-point swing in the other direction. At S=10, penalty die drops P(regular) from 10/100 to 1/100 (9 points); at S=90, from 90/100 to 81/100 (also 9 points). The mechanic is symmetric around S=50 because the canonical d100 distribution is symmetric around the cutoff at the midpoint.
What about the critical (01) and fumble (96–100) bands?: Bonus die makes critical (roll = 01) MORE likely than 1/100, because either of the two tens rolls landing on 0 with units = 1 yields result = 1. P(critical | bonus) = P(U=1 ∧ (T1=0 ∨ T2=0)) = (1/10) · (19/100) = 19/1000. Penalty die makes critical LESS likely — only when BOTH tens are 0 with units=1, which is 1/1000 (1/10 the plain rate). Fumble at high skill (S ≥ 50, fumble = roll = 100 only): bonus die drops fumble from 1/100 to 1/1000 (10× rarer); penalty die lifts fumble from 1/100 to 19/1000 (≈1.9%, the same asymmetry as the critical case but in the opposite tail). The symmetry that holds cleanly for P(regular) doesn't extend to the band edges, where the cutoff is far from the median.