How does this compute exact dice probabilities?

It builds the distribution one die at a time by convolution: it starts with the modifier, then for each die it spreads the current distribution across that die's faces and sums overlapping totals. This gives exact rational probabilities, not a simulation.

What notation can I use?

Standard RPG notation: NdM for N dice of M sides, joined with plus or minus, plus an optional flat modifier. Examples are 3d6, 2d8+1d4, 4d6-2, and 1d20+5. Spaces are ignored and dice may be listed in any order.

Why is the middle total most likely?

Summing several dice concentrates results toward the average because there are many more combinations that produce a middle total than an extreme one. With 3d6 there is only one way to roll 3 but 27 ways to roll a 10 or 11, so the distribution peaks in the centre.

What do at-least and at-most odds mean?

At-least is the cumulative probability of rolling that total or higher, which answers questions like the chance to beat a target number. At-most is the probability of that total or lower. Both are shown for every total so you can read success odds directly.

Is there a limit on pool size?

The tool caps the number of dice and total face range to keep the table fast and readable in the browser. Very large pools (hundreds of dice) are rejected, but any realistic RPG pool computes instantly because convolution is efficient.

What is the Custom Dice Pool Probability Table?

Enter any dice pool such as 3d6+2 or 2d8+1d4 to generate the exact probability distribution table and a text histogram using combinatorial convolution. Shows mean, mode, and at-least / at-most odds. For tabletop RPG designers and players. It runs free in your browser on Gera Tools, with nothing uploaded.

Custom Dice Pool Probability Table

Name: Custom Dice Pool Probability Table
Creator: Gera Tools
License: https://creativecommons.org/licenses/by/4.0/

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This tool gives the exact probability of every possible total for any dice pool, using combinatorial math rather than simulation. Enter notation like 3d6+2 or 2d8+1d4 and read off the full distribution, the average, and cumulative success odds.

How it works

The distribution is built by convolution, one die at a time. Each die spreads the running distribution across its faces and re-sums the overlaps:

start: probability 1.0 at the flat modifier
for each die dM:
    new[t + face] += old[t] / M   for face = 1..M

Because every face is equally likely and independent, this produces the exact probability of each total. The chance of any total is the count of dice combinations that sum to it divided by the total number of combinations.

The 3d6 distribution as a reference case

The classic ability-score roll of 3d6 has 216 equally likely outcomes (6³). Their distribution is:

The minimum total is 3 (all ones) — 1 combination out of 216, probability ≈ 0.5%.
The maximum total is 18 (all sixes) — also 1 combination, probability ≈ 0.5%.
The most likely totals are 10 and 11 — each has 27 combinations out of 216, probability ≈ 12.5%.
Rolling 15 or higher (the threshold that matters for strong ability scores) has a cumulative probability of about 9.3%.

The distribution is symmetric and bell-shaped, which is why 3d6 replaced 1d20 for ability scores in some systems: it concentrates results in the middle and makes extreme rolls rare rather than equally likely.

Using the probability table to design game mechanics

The at-least and at-most columns are the tools that game designers reach for most often:

Setting a target number (TN): If you want about a 50% base success rate, find the median of your dice expression and set the TN there. For 2d6, the median is 7, and the chance of rolling 7 or more is exactly 58.3%.

Comparing pool sizes: Run the same base expression (for example 1d6 vs 2d6 vs 3d6) to see how adding dice changes the distribution. More dice concentrates results toward the middle; fewer dice keeps the distribution flat and high-variance.

Checking modifier impact: Add a flat bonus (3d6+2) and compare the at-least column for the same target number. A +2 on a 3d6 pool shifts the curve meaningfully — the table shows exactly by how much.

Mixed pools for advantage/disadvantage effects: 2d8+1d4 is a common pattern in custom systems. Convolution gives the exact distribution, unlike simulation which requires thousands of rolls to approach it.

Practical tips

Keep your pool under about 20 dice for near-instant browser computation.
Use a flat modifier (the +N part) to shift the entire distribution up or down without changing its shape.
Compare the mean against the mode: if they differ significantly, your distribution is skewed and the most likely result is not the average result.
The at-most column is useful for “roll under” systems where a lower total is better.