Enter the number of dice, the number of faces per die, and a target value to get the exact probability, the full probability distribution, and the mathematical expectation for that roll.
Full probability distribution
This distribution assumes fair, independent dice: every face has an equal chance of landing and one roll does not influence another.
How the probability distribution is calculated
The calculator builds the distribution of possible sums by convolution: it starts with the equal-chance distribution of a single die, then combines it with itself one extra die at a time. For two six-sided dice, this produces the well-known triangular shape peaking at 7, since seven can be formed six different ways (1+6, 2+5, 3+4, 4+3, 5+2, 6+1) out of 36 equally likely combinations, a probability of 6/36, roughly 16.67 percent. Every additional die repeats the same combination step, which is why the same method works for 2d6, 3d8, or 1d20 without any shortcut or approximation.
Abraham de Moivre and the London roots of dice probability
Much of the mathematics behind this calculator traces back to Abraham de Moivre, a mathematician who settled in London after fleeing religious persecution in France and published The Doctrine of Chances in 1718, one of the first books to set out a general method for working out the odds of combined dice throws rather than relying on tables built by trial and error. Gambling houses of the period paid close attention to his results, since a miscalculated dice game could bankrupt the house rather than the player.
Dice pools in British tabletop gaming
Nottingham-based Games Workshop built its Warhammer and Warhammer 40,000 combat systems around pools of six-sided dice, where a single attack can mean rolling a dozen or more d6 at once and counting the successes, a very different question from the single-sum probability this tool answers but built from the same underlying dice mathematics. Role-playing games using the d20 system, widely played across the UK gaming scene, rely on a single twenty-sided die for most actions while damage is often the sum of several smaller dice, exactly the kind of total this calculator models.
What this tool does not cover
It assumes standard, fair dice with no weighting toward any face, and treats each die as fully independent, so it does not model reroll mechanics, exploding dice, advantage or disadvantage rules, or dice pools that only count successes above a threshold, all of which need a different calculation method.