dice-probability

Dice Probability Skill

When to Use

Activate this skill when a user asks about:

• "What are the odds of...", "how likely is...", "what's the probability..."
• "Which is better...", "should I take advantage or +2..."
• "Expected value of...", "average roll for..."
• Comparing dice strategies or builds
• Optimizing roll choices for a game system

Analysis Approach

Small Spaces (exact math)

For rolls with manageable outcome spaces (up to ~6 dice, up to 20 sides, no complex modifiers), enumerate all outcomes exactly.

Uniform (1dN): Every face is equally likely. P(X) = 1/N. Expected value = (N+1)/2.

Sum of NdS: Total outcomes = S^N. Enumerate or use convolution. Approximately normal for large N. Expected value = N * (S+1) / 2. Variance = N * (S^2 - 1) / 12.

Complex Modifiers (simulation)

For rerolls, exploding dice, large pools, or combined modifiers, use Monte Carlo simulation. Run 100,000+ iterations for stable results:

import { roll } from '@randsum/roller'

const N = 100_000
const results = Array.from({ length: N }, () => roll('4d6L').total)
const avg = results.reduce((a, b) => a + b) / N

For reproducible simulations, use a seeded random function:

import { roll } from '@randsum/roller'
import { createSeededRandom } from '@randsum/test-utils'

const config = { randomFn: createSeededRandom(42) }
const results = Array.from({ length: 100_000 }, () =>
  roll('4d6L', config).total
)

When to Use Which

Scenario	Method
1dN, NdS sums	Exact math
Drop/keep (small pools)	Exact enumeration
Rerolls, exploding, caps	Monte Carlo simulation
Comparing two strategies	Both -- exact if possible, simulation otherwise

Common Distributions

Single Die (1dN)

Uniform distribution. E = (N+1)/2, Var = (N^2 - 1)/12.

Die	Expected	Std Dev
1d4	2.5	1.12
1d6	3.5	1.71
1d8	4.5	2.29
1d10	5.5	2.87
1d12	6.5	3.45
1d20	10.5	5.77

Sum of Multiple Dice (NdS)

E = N * (S+1) / 2. Distribution approaches normal as N grows.

Roll	Expected	Std Dev
2d6	7.0	2.42
3d6	10.5	2.96
4d6	14.0	3.42
2d10	11.0	4.06
2d12	13.0	4.88

4d6L (D&D Ability Score)

Drop lowest of 4d6. Skewed high. E = 12.24, Mode = 13, Median = 12.

Advantage / Disadvantage (2d20L / 2d20H)

• Advantage (keep highest): E = 13.82, P(nat 20) = 9.75%
• Disadvantage (keep lowest): E = 7.18, P(nat 1) = 9.75%
• Straight 1d20: E = 10.5, P(any face) = 5%

See references/PROBABILITY_TABLES.md for full distribution tables.

Modifier Impact on Distributions

Modifier	Effect on Mean	Effect on Variance
Drop lowest (L)	Increases	Decreases
Drop highest (H)	Decreases	Decreases
Keep highest (K)	Increases	Decreases
Keep lowest (kl)	Decreases	Decreases
Reroll low (R{<N})	Slight increase	Slight decrease
Explode (!)	Increases (extends upper tail)	Increases
Compound (!!)	Same mean shift as explode	Higher variance on individual dice
Cap (C{>N})	Decreases (truncates top)	Decreases
Cap (C{<N})	Increases (truncates bottom)	Decreases

Modifier execution order matters. Cap applies before drop/keep (priority 10 vs 20/21), so 4d6C{>4}L caps first, then drops -- the dropped die is from the capped pool.

Comparison Framework

When users ask "which is better?", compare along three axes:

1. Expected value -- which averages higher?
2. P(success) -- for a given target, which succeeds more often?
3. Consistency -- which has lower variance / tighter spread?

Present results as: "Strategy A averages X (std dev Y) vs Strategy B averaging X' (std dev Y'). Against DC 15, A succeeds Z% of the time vs B at Z'%."

Example -- advantage vs flat +2 on 1d20:

• 2d20L (advantage): E = 13.82. P(>=15) = 51.0%
• 1d20+2: E = 12.5. P(>=15) = 40.0%
• Advantage is better at mid-range DCs. The +2 bonus is better at extreme DCs (very low or very high targets).

Game System Probabilities

D&D 5e

• P(hit) = (21 - AC + modifier) / 20, clamped to [0.05, 0.95] (nat 1 always misses, nat 20 always hits)
• Advantage impact: roughly equivalent to +3.3 to the roll on average, but non-linear -- strongest near 50% base chance
• 4d6L ability scores: see tables in references

Blades in the Dark (Nd6, keep highest)

Pool	P(Critical)	P(Success)	P(Partial)	P(Failure)
0 (2d6kl)	0%	2.8%	25.0%	72.2%
1d6	0%	16.7%	33.3%	50.0%
2d6	2.8%	25.0%	44.4%	27.8%
3d6	7.4%	29.6%	44.4%	18.5%
4d6	13.2%	31.6%	39.9%	15.3%

Critical = two or more 6s. Success = highest is 6. Partial = highest is 4-5. Failure = highest is 1-3.

PbtA / Root RPG (2d6 + stat)

Stat	P(10+) Strong	P(7-9) Weak	P(6-) Miss
-1	8.3%	33.3%	58.3%
+0	16.7%	41.7%	41.7%
+1	27.8%	44.4%	27.8%
+2	41.7%	41.7%	16.7%
+3	58.3%	33.3%	8.3%

Salvage Union (1d20, roll under target)

P(success) = target / 20. Critical success on 1 (5%). Critical failure on 20 (5%).

References

• Full probability tables -- 4d6L distribution, advantage/disadvantage curves, Blades and PbtA lookup tables
• RANDSUM Dice Notation Specification -- complete modifier syntax and taxonomy
• Game system mechanics -- how each game interprets rolls