Dice Probability Calculator

What this generator does

Produces the complete probability distribution for the total of any number of identical dice: every reachable total, the exact count of ways to reach it, its probability as a fraction, and the cumulative chance of rolling that total or higher. It is built by convolution rather than simulation, so the counts are exact outcome counts rather than estimates from a sample — 'six ways in thirty-six', not 'about 16.7% over a million rolls'.

How to use this tool

1. Choose how many dice and how many sides each has.
2. Add a modifier if your game applies one to the total.
3. Press Calculate.
4. Read the distribution: every possible total, how many ways it happens, and the odds of rolling it or better.

Understanding the controls

Number of dice

How many are rolled and summed. This is the control that changes the shape: one die is flat, two form a triangle, and from three onwards the distribution starts to look like a bell, with extremes becoming rapidly less likely.

Sides per die

Standard dice are six-sided but the arithmetic is identical for four, eight, ten, twelve or twenty. Larger dice widen the range without changing the shape.

Modifier

A fixed number added to the total. It shifts every outcome along without changing any probability, which is exactly why a flat bonus is a much weaker change than adding another die.

Worked examples

Two six-sided dice

7 comes up 6 ways in 36. Twelve comes up 1 way in 36 — six times rarer.

Three six-sided dice

10 and 11 are the most likely totals, at 27 ways each out of 216.

Why more dice is not a bigger bonus

4d6 ranges 4–24 but two-thirds of rolls land between 11 and 17.

One d20

Every value from 1 to 20 is equally likely — a flat 1 in 20, which is why a single die feels so swingy.

Common use cases

• Working out whether a tabletop check is worth attempting before rolling
• Designing a game's difficulty numbers against what the dice actually do
• Teaching probability distributions with dice, where the counting can be verified by hand
• Settling arguments about which totals are common on three or more dice
• Understanding why more dice makes results cluster rather than spread

How this generator works

The distribution is built one die at a time. Start with a single outcome of zero, then for each die replace every running total with all the totals that die could produce from it, adding up the ways each is reached. After all the dice are folded in, the table holds the exact number of ordered outcomes that reach every total, and those counts sum to sides^dice — a property the tool checks rather than assumes. Cumulative odds are then a running sum from the top down. Nothing is sampled at any point: a simulation of a million rolls would give an approximation of numbers that can be counted exactly in a few milliseconds.

Randomness and fairness

Nothing is rolled and nothing is random. The whole table is derived arithmetic, so the same inputs always produce the same distribution — which is the point of a calculator as opposed to a roller.

For how randomness is produced across the whole site, see how Generate Random works.

Assumptions this tool makes

• Every face is equally likely and each die is independent of the others, which is what a fair die means.
• Dice are summed. Systems that take the highest, lowest or a count of successes are outside this model.

Limitations and good to know

• It handles identical dice summed together. Mixed pools, exploding dice, and reroll or keep-highest mechanics are different distributions and are not modelled here.
• It gives the distribution of the total, not of individual dice. The chance of rolling at least one six on four dice is a different question from the chance of the total exceeding a number.
• Very large pools are refused rather than shown, because a table of thousands of rows is not something anybody reads.
• It says nothing about whether physical dice are fair. It describes ideal dice; a chipped or loaded die does not follow this table.

Common mistakes

Assuming every total is equally likely

Only a single die is flat. As soon as dice are summed the middle totals have many more ways to occur — seven has six on two dice, twelve has one.

Treating a +1 modifier as roughly the same as an extra die

A modifier shifts the whole distribution by one and changes no probability. An extra die changes the shape, the range and the variance. They are not comparable adjustments.

Reading the individual probability when the cumulative one is wanted

Most game questions are 'do I roll at least N', which is the cumulative column. The per-total column answers the narrower question of hitting exactly that number.

Practical tips

• For game design, set target numbers against the cumulative column. It tells you directly what proportion of attempts succeed, which is the number that determines how a game feels.
• Compare 3d6 with 1d18. They cover almost the same range and behave completely differently — one clusters hard around the middle, the other is flat.
• If you are teaching this, have students enumerate two dice by hand first. Thirty-six outcomes is small enough to write out, and it makes every later case obvious.

Troubleshooting

My tabletop system's odds do not match

Check whether it uses advantage, exploding dice or a success-counting pool. Those are different mechanics, not different numbers, and a sum distribution does not describe them.

The table is refused as too large

Reduce the number of dice or the sides. A distribution with thousands of rows is arithmetically fine and practically unusable, so it is declined rather than shown.

Privacy and your data

Only dice counts and sides are entered, and they stay in your browser. Nothing is stored or uploaded, nothing is written into the page address, and analytics records only that the calculator was used.

Frequently asked questions

Is this simulated?

No. Every figure is an exact count of outcomes, produced by folding one die into the distribution at a time. A simulation would approximate numbers that can simply be counted, and the counts are checked to sum to the total number of possible rolls.

Why does seven come up most on two dice?

Because it has the most ways to happen: 1+6, 2+5, 3+4, 4+3, 5+2 and 6+1 — six ordered outcomes out of thirty-six. Twelve has only 6+6. The totals in the middle simply have more combinations that reach them.

Does this work for twenty-sided dice?

Yes, for any number of sides from two upwards. The arithmetic is identical; only the range changes.