Dice Probability Studio
Compute exact probability distributions for any multi-dice pool from d4 to d100. Includes exploding dice, D&D 5e advantage/disadvantage, and a side-by-side comparison chart across dice types.
Setup
Distribution
Probability table
| Sum | Probability | % | Cumulative ≤ |
|---|
Compare dice types (single dN, mean & spread)
Each bar shows the standard deviation of one fair die. Bigger die ≠ always better — it means wider swing per roll.
What this tool actually solves
You have a dice pool — say 3d6 for an ability score, or 8d6 for Shadowrun damage — and you need to know what can actually happen. Average damage is a single number; the distribution tells you the chance of a crit, the chance of a whiff, the median, and the long tails. Most online dice calculators stop at the average.
We compute the full exact distribution using convolution (which is the textbook way and produces integer-exact probabilities), then draw it so you can see the shape — symmetric for raw sums, skewed when exploding dice are on.
Who typically uses this
- D&D players checking whether "advantage" really gives a +5 swing or "just feels good".
- Game designers balancing weapon tables in homebrew systems.
- Shadowrun / World of Darkness players computing hit probabilities with exploding dice pools.
- Teachers illustrating the central-limit theorem visually with dice sums.
How to read the output
Look at three numbers: the mode (most likely outcome, usually near the middle), the 5th–95th percentile band (where 90% of rolls land), and the probability of beating your target. For 3d6 the mode is 10–11, the 5–95 band is roughly 6–15, and the average is 10.5. For 8d6 the mode is around 27–28 with a much tighter 90% band.
Honest limitations
- Exploding dice creates an open-ended distribution; we cap re-rolls at 30 to avoid an infinite tail (probability past 30 explosions is below 1 in 10 million for most dice).
- "Advantage" here assumes two independent identical dice — true for physical dice, not for online RNGs with shared entropy.
- This tool assumes fair dice. Loaded dice (cheating) are not modeled.
Frequently asked questions
Is exploding a d6 the same as rolling 2d6 and adding?
No. Exploding a single d6 on max gives average ≈ 7, vs. 2d6 average 7 but with a triangular (not skewed) distribution. Exploding adds a long right tail.
Why does d20 advantage feel weaker than +5 sometimes?
Because the median shift of advantage is about +3.3, not +5. The +5 is a mean shift; the difference is in the tails — advantage removes the worst rolls (no more natural 1s on a single d20) more than it boosts the median.