Dice Probability Calculator
Calculate the probability of rolling a given total with several dice. Shows the exact probability of a target sum, the at-least and at-most probabilities, expected value, variance, and a bar chart of the distribution.
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Input
Enter the number of dice, the number of faces, and a target sum to find the probability that the dice add up to that value.
dice
faces
sum
The probability that the dice add up to this value.
Result
Probability of a total of 7
16.67percent
6 ways out of 36
At least 7
58.33 percent
At most 7
58.33 percent
Expected value
7
Variance
5.833
Standard deviation
2.415
Total outcomes
36
Distribution of the total
How it works
- The distribution of the sum of n dice with m faces (each face from 1 to m equally likely) is computed exactly by convolution.
- The total can range from n, when every die shows 1, up to n times m, when every die shows m.
- The probability of a target sum equals the number of ways to reach that sum divided by the total number of outcomes, m to the power n.
- The at-least probability adds the outcomes for every sum greater than or equal to the target, and the at-most probability adds those less than or equal to it.
- Each single die has expected value (m plus 1) divided by 2 and variance (m squared minus 1) divided by 12, so the total for n dice is n times these values.
Frequently asked questions
- How is the probability of a specific total calculated?
- It is the number of ways to make that total ÷ the total number of outcomes (m to the power n). The distribution for n dice with m faces is built by convolution, so the probability is exact rather than an approximation.
- What range of totals is possible, and what are the at-least and at-most figures?
- The sum ranges from n (all dice showing 1) up to n × m (all showing m). The at-least probability adds up the cases at or above your value, and the at-most probability adds up the cases at or below it.
- How are the expected value and variance found?
- For one die the expected value is (m + 1) ÷ 2 and the variance is (m² − 1) ÷ 12. For n dice both are multiplied by n — for example, a single 6-sided die has an expected value of 3.5.
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