Dice Probability Calculator

About this tool

Two six-sided dice do not produce eleven equally likely totals. Seven comes up six times as often as two, because there are six ways to make seven and only one way to make two. That single fact is what a distribution table is for, and it is why "just divide by the number of outcomes" gives the wrong answer to almost every practical question at a table.

The shape gets more surprising as dice are added. One die is flat: every face is equally likely. Two dice form a triangle. Three dice already look like a bell, and by four the middle is so dominant that the extremes are close to irrelevant. The calculator builds this by convolving one die at a time, which is the exact answer rather than an approximation from random rolls.

Mixed pools work the same way. A 2d6+1d8 pool has a range of 3 to 20, but its peak sits at 14 and 15 with a small dip between them, and nothing about the individual dice would tell you that. Add a flat modifier and the whole curve slides sideways without changing shape — a +3 on 2d6 does not make high results more likely, it just moves the target.

Why two dice make a triangle and not a flat line

There are 36 equally likely pairs from two six-sided dice, and the totals are not spread evenly across them. Seven has six ways, two and twelve have one each. Sorted by total, the counts rise to a peak and fall away, forming a triangle.

The counting is worth doing once by hand. To make 2 you need 1+1, one way. To make 3 you need 1+2 or 2+1, two ways. The count climbs by one for each total until 7, where the six pairs are 1+6 through 6+1. After that it mirrors: 8 has five ways, and 12 has one. Thirty-six outcomes in total, and the middle claims six of them.

The triangle becomes a bell as dice are added because each new die convolves the existing shape with a flat one. Convolution has a smoothing effect: sharp corners round off and the middle fills in. Three dice already look curved; four are close to a normal distribution. That is also why the standard deviation grows with the square root of the number of dice rather than linearly — doubling the dice does not double the spread.

The practical consequence is that the extremes become very unlikely very fast. On 3d6 the chance of an 18 is one in 216, while the chance of 10 or 11 together is more than a quarter. Any rule that hinges on an extreme total is far harsher than the range alone suggests.

Mixing dice sizes without flattening the curve

A 2d6+1d8 pool runs from 3 to 20 and peaks at 14 and 15, with a slight dip between them. You cannot get that shape by treating the pool as one die with an average number of sides, or by averaging two curves — only by convolving each group.

Convolution means: for every total the pool can already reach, spread its probability evenly over the new die faces. Do that once per die and the result is exact. It also explains the small dip at the top of a mixed pool: the d8 contributes a flat top, and the d6 contributes a peak, so their sum has two nearly equal maxima with a shallow valley between.

The expected value is simply the sum of the individual expectations — 7 for 2d6, 4.5 for 1d8, so 11.5, plus any flat modifier. Variances add too, which means a high-variance die makes the pool more swingy regardless of where it sits in the expression. Order does not matter anywhere in this calculation.

The limits in the tool are practical rather than mathematical. Thirty dice across all groups and no more than a three-thousand-wide span keeps the computation instant and keeps the table readable. Beyond that you are usually modelling something that would be resolved by an app or a simulation anyway.

At least, at most, exactly — three different questions

On 2d6, the chance of at least 10 and the chance of exactly 10 are 16.7% and 8.3%. Both are correct; they answer different questions, and a rulebook almost always means the first one.

A check that succeeds on a total of 10 or more is asking for the cumulative chance, which on 2d6 is the sum of the 10, 11 and 12 rows. Reading the single-row figure instead halves the answer, and the error grows as the target moves toward the tail. At 12 or higher the cumulative chance is 2.8% and the single-row chance is also 2.8%, so a spot check at the extreme would not reveal the mistake.

The reverse question matters just as much. Some rules fail on a high roll, or ask for "10 or under", and then the cumulative should be read from the bottom of the table instead of the top. The calculator keeps three columns precisely so that the direction of the comparison is never something the reader has to work out.

One useful pattern: the cumulative column changes slowly in the middle of a bell and fast at the edges. Moving a target by one point near the peak shifts the odds by a few percent; moving it by one point near the tail can halve them. That is the intuition behind bounded accuracy systems, and it falls straight out of the table.

Why floating-point probabilities instead of combinations

Binomial coefficients are elegant for a single probability, but they leave the safe integer range as parameters grow, and a question like at least one six still needs care over double counting. This page accumulates floating-point probabilities instead, so the edges stay right.

Two d6 produce a triangle: seven appears 6 times in 36, or 16.67%, while two and twelve appear once each, or 2.78%. Hand-computing at least one six invites writing 6/36, whereas the answer is 11/36, or 30.56% — because double six was counted twice.

Coefficients overflow at scale: twenty d20 already exceed the integers that can be represented exactly, while a floating-point recurrence keeps every step between 0 and 1, so nothing overflows. The cost is a tiny error in the last digits, which cannot flip a comparison against zero.

At least, at most and exactly are three separate questions with three separate computations. The chance of exactly twelve is not the chance of at least twelve minus the chance of at least thirteen, and intuition usually fails here, so each is computed in its own right and marked on the distribution axis.

Unsupported notation is refused rather than guessed: 2d, d and 3d6+ all raise errors. Guessing one symbol wrong makes every number after it wrong, and the reader cannot see it from the result.