How this calculator works
A fair s-sided die assigns probability 1/s to each integer face from 1 to s. Adding a die convolves its face probabilities with the existing sum distribution. Repeating this produces the exact discrete model, evaluated with floating-point arithmetic. Sum outcomes are not equally likely: middle totals generally have more contributing face combinations. The maximum supported sum is 500 so the complete distribution fits the table.
Formula
P_d(t) = sum from j = 1 to s of P_(d - 1)(t - j)/s, starting with P_0(0) = 1. Expected sum = d(s + 1)/2.
Worked example
Two six-sided dice sum to seven in six ordered ways: (1,6) through (6,1). Each ordered pair has probability 1/36, giving 6/36 = 1/6.
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Frequently asked questions
Are all sums equally likely?
No. For two six-sided dice, sum two has one ordered outcome while sum seven has six.
Are the target inequalities inclusive?
Yes. At least and at most both include the entered target.
Can this model loaded or mixed dice?
This page assumes identical fair dice. The dice average calculator supports different side counts for expected totals, but does not infer loaded-face probabilities.