Dice Probability Calculator

The calculator builds the full distribution of the sum by combining each die's faces. For n fair dice with s sides, there are s to the power of n equally likely ordered outcomes, so the probability of a given sum is the number of ways to roll that sum divided by s to the n. For example, two six-sided dice have 36 outcomes and six of them total 7, so P(sum = 7) is 6 over 36, or about 16.67 percent.

The most likely total sits at the center of the range. For dice with the same number of sides, the peak is around n times (s + 1) divided by 2, which is also the expected value. With two six-sided dice the most likely sum is 7. The calculator marks this value as the mode and shows its probability.

At least gives P(sum >= target), the chance the total is the target or higher. At most gives P(sum <= target), the chance the total is the target or lower. Greater than and less than exclude the target itself. The calculator returns all five comparisons at once so you can read whichever you need.

The expected value of the sum is n times (s + 1) divided by 2, where n is the number of dice and s is the sides per die. The variance is n times (s squared minus 1) divided by 12, and the standard deviation is the square root of that variance. For two six-sided dice the mean is 7 and the standard deviation is about 2.42.

Yes. You can set any number of sides from 2 to 100, so it handles four-sided, eight-sided, ten-sided, twelve-sided, twenty-sided, and other polyhedral dice common in tabletop games. You can also roll up to 50 dice at once. The full sum distribution and summary statistics update for whatever combination you enter.

The probabilities are exact for fair, independent dice, computed from the complete distribution rather than a simulation. The number of ways for each sum is shown as a whole number whenever the total outcome count (s to the n) fits in a safe integer. For very large dice counts that total is too big to display exactly, so the calculator shows probabilities only while keeping them precise.