Dice sum probability

When you roll several fair dice and care about the total, every possible sum has its own exact probability. Two standard six-sided dice make seven more likely than two or twelve; three dice shift the mass toward the middle again; custom polyhedral dice change the range entirely. This calculator takes the number of dice k and the number of faces s on each die, expands the generating function (x + x² + … + x^s)^k with deterministic integer arithmetic, and returns the full probability mass function: ways, reduced fraction, and decimal probability for every reachable total from k through k·s. Values of k below one or s below two are rejected with a clear validation error so malformed game rules never produce a silent nonsense table.

What dice-sum probability means and why generating functions help

A single fair s-sided die shows each face from one through s with equal chance 1/s. When you roll k independent copies and add the faces, the sample space has size s^k equally likely outcomes. The sum ranges from the absolute minimum k (every die shows one) to the absolute maximum k·s (every die shows its highest face).

Intermediate totals are not equally likely: many different face combinations can produce the same middle sum, while the extreme totals have only a few microstates. The classical tool for counting those combinations is the generating function whose exponents track the sum and whose coefficients count the ways. One die contributes the polynomial x + x² + … + x^s.

Rolling k independent dice multiplies that polynomial by itself k times, so the coefficient of x^t in (x + x² + … + x^s)^k is exactly the number of ordered outcomes whose faces add to t. Divide that coefficient by s^k and you obtain the exact probability of total t. This is the same idea behind dice tables in tabletop RPGs, board-game balance sheets, and introductory discrete-probability courses.

It is also fully deterministic: given the same k and s, every correct implementation must produce the same ways, the same fractions, and the same ordered list of totals, which makes the distribution safe to pin in tests and teaching materials without Monte Carlo noise.

How this calculator expands the generating function

Call the capability with two required integers: k, the number of dice, and s, the number of faces on each die. The engine validates that k is an integer from one through forty and that s is an integer from two through one hundred. After validation, the core starts from the trivial generating function for zero dice (coefficient one at sum zero) and folds in one die at a time by convolving the current coefficient array with the face vector of length s.

Each convolution is exact BigInt arithmetic, so large way counts never round early. When all k dice have been folded in, the engine reads coefficients from sum k through sum k·s, checks that they add to s^k, reduces each ways/total fraction with a greatest-common-divisor step, and emits a stable floating probability for display.