| name | dice-probability |
| description | Analyze probability distributions for RANDSUM dice notation. Use when users ask about odds, expected values, probability of specific outcomes, or want to compare different roll strategies. Handles standard dice, drop/keep, advantage/disadvantage, rerolls, exploding dice, and game-system outcomes. |
| license | MIT |
| metadata | {"author":"RANDSUM","version":"1.0","repository":"https://github.com/RANDSUM/randsum"} |
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:
- Expected value -- which averages higher?
- P(success) -- for a given target, which succeeds more often?
- 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
