P ProbLab
Tools/Dice Probability Studio

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

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Distribution

Probability table

SumProbability%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.

Frequently asked questions

How do I use Dice Probability Studio step by step without mistakes?
Follow the order given in the usage section: Compute exact probability distributions for any multi-dice pool (d4–d100). Includes exploding dice, D&D 5e advantage/disadvantage, critical-hit detection, and a side-by-side comparison chart. The step-by-step part matters less than the unit check that sits inside it. Confirm every unit matches the thing you are actually trying to measure before you press calculate - that single check catches more wrong answers than any other habit, and it costs about ten seconds.
What are the common mistakes when using Dice Probability Studio?
Most common mistakes come down to four habits: entering 20 when the field expects 0.2 because the field is a percentage; leaving a field blank and accepting the default without checking what it stands for; changing one input and reading a result that was calculated before the change; and quoting the output as an exact figure when the page describes it as a reference estimate. Press calculate again after any edit, and treat the number as a reference point rather than a final answer.
How do I read the result from Dice Probability Studio?
Read the final figure first, then come back to this section for the conditions behind it: 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 Keep two things apart: the number itself, which comes from the values you typed, and the assumptions it depends on, which decide whether that number is meaningful. The page sets out what a higher or lower result means for your case, so check that direction as well as the magnitude.
Can I act on the Dice Probability Studio result directly?
Treat it as one reference sheet before you decide, rather than as the answer itself: Exploding dice creates an open-ended distribution; The output describes a magnitude under the conditions you entered. It does not include what it actually costs to act, what might change while you act, or how the other side behaves. Before you commit, compare the result against reality and note which assumptions do not hold where you are.