Dice Probability Matrix
Not just "what's the chance". For any polyhedral dice stack (e.g. 3d6, 4d8, 2d20+3) we give you the full probability mass function, mean, variance, skewness and the chance to beat or fall below any threshold — useful for tabletop RPG designers, boardgame playtesting and Craps players who care about the real curve.
Adds per-face bias to test "loaded" dice. Use sparingly.
| Sum | Combinations | Probability | Cumulative ≤ |
|---|
How the math works
The number of ordered outcomes for N dice of F faces is F^N. The probability mass function (PMF) is computed by enumerating all outcomes and counting how many produce each sum, optionally weighted by a per-face bias. For dice up to F=10 and N=10 we enumerate directly; for larger stacks we use a generating-function convolution in O(N · F²) time which keeps it instant in the browser.
- Mean:
(N·(F+1))/2 + mod - Variance:
N·(F²−1)/12 - Skewness: 0 for uniform dice; bias shifts it
- Central Limit: as N grows the sum approaches N(μ,σ²)
What this is good for
Tabletop RPG / boardgame
Validate damage curves for "greatsword 2d6 + Str" or random encounter tables. Quickly check whether your homebrew feat is over- or under-powered compared to the baseline 1d20 distribution.
Stats teaching
Demonstrates the Central Limit Theorem live: the histogram visibly approaches a bell curve as N grows. Pair it with a moment estimate to show skewness vanishing.
How to read the chart
Each bar represents one possible sum; the height is its probability. Hover the table row for the same value in percent form. The cumulative column is "P(X ≤ sum)" — useful when you want the chance of staying under or over a threshold, and a foundation for one-sided tests.
When not to trust this
- Real dice are not perfectly uniform — printed d6s can skew by ±5%. The bias input tries to capture this but it is a coarse knob.
- Some rules (advantage/disadvantage, exploding dice) involve dependent rolls. Use a custom distribution in those cases.
- If you are betting money and need exact Craps odds, the official Wizard of Odds tables are more authoritative than any web calculator.
FAQ
Why is the histogram not bell-shaped for small N?
For small dice counts the probability mass function can take a triangular shape (3d6) or stay flat (uniform). The shape converges to a bell curve (normal distribution) as N grows — that convergence is the Central Limit Theorem.
Can I model "exploding" dice (e.g. Shadowrun)?
Not directly here. Exploding dice involve dependent rolls and the distribution has heavy tails. If you need that, the cleanest workaround is to simulate with the same dice, increase the dice count, or set a very high flat target.
Does the "bias" input make my game balance look broken?
It is a uniform per-face adjustment in probability, not a per-face specific shift. Real "loaded" dice favour specific face values asymmetrically. Use this only to understand how sensitive the chart is to bias, not to recreate a specific die.
Full enumeration for poker hands.
Bust probability by dealer upcard.
Without-replacement sampling.