Dice Probability Calculator

Rolling dice is one of the simplest and most intuitive ways to explore probability. With a single fair six-sided die, every face has equal probability: 1/6 or about 16.67%. When you add a second die and look at the sum, the probabilities are no longer equal: some sums like 7 can be made in many more ways than extreme values like 2 or 12. This calculator works out the exact probability distribution for any combination of identical dice with any number of sides, from the classic d6 up to d20 and beyond, for up to six dice at once. The distribution is computed by convolution: starting from the uniform distribution for one die and repeatedly folding in each additional die. The result is the number of ways to achieve each possible sum out of the total number of outcomes (the number of faces raised to the power of the number of dice). You choose how many dice, how many sides, and which specific sum you want to target, and the calculator returns the exact probability as a fraction and as a percentage, the most probable sum, the expected value, and a full distribution table showing the probability of every possible sum. When rolling two six-sided dice (2d6), a sum of 7 can be made in 6 out of 36 ways, giving a probability of 16.67%. The tool suits students learning probability, board game enthusiasts assessing the odds of specific outcomes, and game designers balancing mechanics around dice results.

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Number of dice

Sides per die

Target sum

16.67%

P(sum = 7) with 2d6

Exact fraction1/6

Most likely sum7

Expected value7.00

Total outcomes36

How it works

The calculator builds the probability distribution by convolution. It starts with a count array of ones (one way to get each face value on a single die). For each additional die it convolves the current array with the single-die array by iterating over every combination of current sum and new face. The resulting count array at each index divided by the total number of outcomes (sides raised to the number of dice) gives the probability for that sum. The expected value is the sum of each outcome times its probability.

Worked example

Rolling 2d6: total outcomes = 6 x 6 = 36. The sum of 7 can be formed as (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 ways. P(sum = 7) = 6/36 = 16.67%. The most probable sum is 7 and the expected value is (2+3+...+12) weighted by probability = 7.00. These match the default values pre-filled above.

Related calculators

Expected Value Calculator: calculate the long-run average of any probability distribution.
Odds Probability Converter: convert between fractional odds and percentage probability.
Bayes Theorem Calculator: update probabilities using prior and conditional information.
Percent Error Calculator: measure how far a result deviates from an expected value.

Frequently asked questions

What is the probability of rolling a 7 with two dice?

With two standard six-sided dice, a sum of 7 can be made in 6 ways: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). The total number of outcomes is 6 x 6 = 36. So the probability is 6/36 = 1/6, which is approximately 16.67%. Seven is the most likely single sum when rolling two dice.

How do you calculate dice probabilities?

For multiple identical dice, you can build the distribution by convolution. Start with a uniform distribution for one die (each face has probability 1/n). For each additional die, convolve the current distribution with the single-die distribution. The resulting counts for each possible sum, divided by the total number of outcomes (n raised to the power of the number of dice), give the probabilities.

Why is 7 the most common sum for two six-sided dice?

The sum of 7 can be formed in the most ways: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 combinations out of 36. Sums closer to the middle of the possible range always have more combinations than extreme sums, which is why the distribution of two-dice totals is roughly bell-shaped and peaks at 7.