The formula
probability = favourable combinations ÷ sides raised to the number of dice
Where it comes from
With a single die every face is equally likely, but once you add several that stops being true: there is only one way to roll 2 with two dice and six ways to roll 7. The calculation walks through every possible combination, building them die by die, which is the only way to stay exact with many dice.
How to work it out by hand
- Count the possible combinations: sides raised to the number of dice
- Count how many ways the total you want can be made
- Divide the favourable ones by the possible ones
What is worth knowing
The average total always falls in the middle of the range and is the most likely, because it is the one with the most combinations behind it. From three or four dice upwards the distribution looks a great deal like a bell curve, which is a particular case of the central limit theorem. That is why rolling 3d6 in a roleplaying game gives results clustered tightly around 10.5, while 1d20 spreads all twenty results evenly.