Dice probability calculator
Probability of rolling a given sum on several identical dice.
Fill in the fields and the result will appear here automatically.
Calculates the chance of exactly a chosen sum on 1–10 identical fair dice with 2–100 faces numbered from 1 to the number of faces. Rolls are independent, so every ordered outcome has equal probability. Favourable and total counts use exact integer arithmetic; their probability ratio is displayed rounded. Ten hundred-sided dice give 100¹⁰ = 10²⁰ outcomes, exceeding 2⁵³, whereas ten twenty-sided dice give 20¹⁰ = 10,240,000,000,000 and remain below that boundary.
How it works
Formula and logic
Favourable combinations come from inclusion–exclusion over the number of dice that overshoot their maximum. Total outcomes are sides raised to the number of dice, and the probability is their ratio. Small nonzero percentages that would round to 0.00% are shown in scientific notation.
Example
Two six-sided dice make seven in six of thirty-six ways, which is 16.67%.
Fields and units
- Number of dice — unitless
- Sides per die — unitless
- Target sum — unitless
How to use
- — Enter how many dice are rolled.
- — Enter how many sides each die has.
- — Enter the sum you are interested in.
- — The sum must be between the number of dice and dice × sides.
Method and limitations
- Calculation method
- Formula and logic
- Data or methodology source
- OpenStax: outcome-count ratios for equiprobable models and fair dice OpenStax: independent intersection, union and distinction from disjoint events
- Limitation
- Calculates the chance of exactly a chosen sum on 1–10 identical fair dice with 2–100 faces numbered from 1 to the number of faces. Rolls are independent, so every ordered outcome has equal probability.
FAQ
Why is seven the most likely sum on two dice?
Because it has the most combinations: six of them, from 1+6 through 6+1. Two and twelve have one each, which is why they show up six times less often.
Does this cover dice with different numbers of sides?
No, all the dice here are identical. Mixed sets — a d6 with a d8, say — need a different count and are not what this calculator computes.
What is the expected sum?
The average over many rolls: dice × (sides + 1) ÷ 2. For three six-sided dice it is 10.5, which is why ten and eleven are the most common results.
How do I get the chance of at least a given sum?
Add up the probabilities of that sum and every higher one. This calculator answers for one exact sum at a time.