Dice Probability Calculator
Find the exact probability of rolling a chosen sum with any number of dice and any number of sides.
Condition
Sum exactly 7
How does this calculator work?
The probability of a dice total equals the number of face combinations giving that sum divided by s^n. This tool convolves n uniform {1..s} dice to count those combinations, then reports exact, at-least, or at-most probabilities along with the mean n(s+1)/2 and the full distribution.
Formula
How this is calculated
Enter the number of dice (n), the number of sides on each die (s), and a target sum (t). The condition dropdown selects whether you want the sum to be exactly t, at least t, or at most t.
The distribution of the total of n s-sided dice is built by convolving n copies of the uniform distribution over {1, 2, ..., s}. Starting from one die (each face equally likely), each additional die shifts and spreads the running array so that dist[k] holds the number of face combinations that produce sum k. The probability of any exact sum is ways(k) divided by the total number of equally likely outcomes, s^n. "At least" and "at most" probabilities sum the relevant tail of that array.
The smallest possible sum is n (all ones) and the largest is n·s (all maximum faces). The mean total is n(s+1)/2 and the variance is n(s²−1)/12, since each die has mean (s+1)/2 and variance (s²−1)/12 and the dice are independent. All dice are assumed fair and independent; targets outside the n…n·s range simply yield probability zero for "exactly".
Frequently asked questions
There are 6 ways to make 7 out of 36 equally likely outcomes, so the probability is 6/36 = 1/6 ≈ 16.667%. Seven is the most likely sum with two standard dice.
Extreme sums can be formed only one way (all ones or all maximums), while central sums can be formed by many combinations. Convolving the uniform distributions produces a bell-like shape peaking near the mean n(s+1)/2.
Yes. Set sides to any value of 2 or more (for example 20 for a d20), and the calculator builds the full distribution by convolution for any number of dice.
Also known as
TG we-Calculate Editorial Team. (2026). Dice Probability Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/dice-probability-calculator
TG we-Calculate Editorial Team. "Dice Probability Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/dice-probability-calculator.
TG we-Calculate Editorial Team, "Dice Probability Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/dice-probability-calculator
@misc{wecalculate_dice_probability_calculator, title = {Dice Probability Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/dice-probability-calculator}}, year = {2026}, note = {TG we-Calculate} }
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