🎲Dice Probability Calculator
e.g. 6, 8, 12, 20
Dice probability uses combinatorics to calculate the exact likelihood of rolling specific sums or outcomes with one or more dice. The distribution of sums follows a discrete probability distribution that becomes approximately bell-shaped (normal) as more dice are added — an illustration of the Central Limit Theorem.
- 1Total outcomes for d dice with s sides: sᵈ
- 2Count ways to roll exactly target sum T: dynamic programming (convolution of dice distributions)
- 3P(sum = T) = ways / total outcomes
- 4Expected sum for d dice with s sides: d × (s+1)/2
- 5Variance: d × (s²−1)/12
2d6, sum = 7=6/36 = 16.7%Most common sum on 2d6; 6 ways out of 36
2d6, sum = 12=1/36 = 2.8%Only one way: (6,6)
3d6, sum = 10=27/216 = 12.5%Peak of 3d6 distribution
| Sum | Ways | Probability |
|---|---|---|
| 2 | 1 | 2.78% |
| 3 | 2 | 5.56% |
| 6 | 5 | 13.89% |
| 7 | 6 | 16.67% (most likely) |
| 8 | 5 | 13.89% |
| 11 | 2 | 5.56% |
| 12 | 1 | 2.78% |
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