Dice Probability Calculator
Why Seven Comes Up More Than Any Other Roll
Roll two six-sided dice and 7 appears more often than any other total — not because of luck, but because more combinations produce it. There's only one way to roll a 2 (1+1), but six different combinations produce a 7. This calculator counts every possible combination for any number of dice and sides, then computes the exact probability of hitting a specific sum, or a sum at or above/below a target.
The Formula
Ways(sum) = count of dice-roll combinations producing that sum
P(sum = target) = Ways(target) / Total Outcomes
P(sum ≥ target) = Σ Ways(s) for s ≥ target, divided by Total Outcomes
P(sum ≤ target) = Σ Ways(s) for s ≤ target, divided by Total Outcomes
Ways(sum) is computed by exhaustively counting every combination, not estimated — this is exact, not an approximation.
The Classic 2d6 Distribution
| Sum | Combinations | Probability |
|---|---|---|
| 2 | 1 / 36 | 2.78% |
| 3 | 2 / 36 | 5.56% |
| 4 | 3 / 36 | 8.33% |
| 5 | 4 / 36 | 11.11% |
| 6 | 5 / 36 | 13.89% |
| 7 | 6 / 36 | 16.67% |
| 8 | 5 / 36 | 13.89% |
| 9 | 4 / 36 | 11.11% |
| 10 | 3 / 36 | 8.33% |
| 11 | 2 / 36 | 5.56% |
| 12 | 1 / 36 | 2.78% |
This is the exact distribution behind the 6-8 tile placement strategy in Catan and the pass-line odds in craps — both games are built directly around 2d6 probability.
Where This Gets Used
- Game design and balancing — designers use exact dice distributions to set target numbers that hit a desired frequency.
- Craps and casino games — the pass-line bet's house edge is derived directly from 2d6 sum probabilities.
- Tabletop RPG mechanics — systems using multiple dice (2d6, 3d6, 4d6-drop-lowest) rely on the same combinatorics to set difficulty thresholds.
How to Use This Calculator
- Choose Exact Sum, At Least Sum, or At Most Sum.
- Enter the number of dice being rolled.
- Enter the number of sides per die.
- Enter the target sum.
- Select Calculate to see the exact probability.
Related Calculations
Check resource-tile odds built on the same 2d6 distribution with the Catan Resource Probability Calculator, or compare combat odds with the Risk Battle Odds Calculator.