Dice Probability Calculator (Sum of Several Dice)
Choose the dice and the total you are interested in.
Your numbers
Your results
Probability of exactly this total
Probability of exactly this total
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Probability of at least this total
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Probability of at most this total
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Exactly: about 1 in
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Possible totals
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Average total
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Dice Probability Calculator (Sum of Several Dice) by TheCalcTools
How the Dice Probability Calculator (Sum of Several Dice) works
Each of the mⁿ outcomes is equally likely. The number of ways to reach a total at most s comes from an inclusion–exclusion formula: Σ (−1)^k C(n, k) C(s − mk, n), so no simulation is needed.
Totals near the middle are most likely; with more dice the distribution approaches a bell curve.
The formula
ways(sum ≤ s) = Σₖ (−1)^k C(n, k) C(s − mk, n)
Worked example
Using the default values shown in the calculator:
| Number of dice | 3 |
|---|---|
| Sides per die | d6 |
| Target total | 10 |
- Three d6 totalling exactly 10: 12.50.
- At least 10: 62.50.
Result:
| Probability of exactly this total | 12.50% |
|---|---|
| Probability of at least this total | 62.50% |
| Probability of at most this total | 50.00% |
| Exactly: about 1 in | 8.0 |
| Possible totals | 3 to 18 |
| Average total | 10.5 |
Frequently asked questions
What is the most likely total with two dice?
7, with a probability of 1/6.
Chance of 15 or more with 3d6?
About 9.26%.
Sources and further reading
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This calculator gives estimates for information only. Check important results independently before you rely on them. See our disclaimer.