Dice probability calculator
Calculate the probability of any dice roll outcome. Choose the number of dice, set your target, and see exact probabilities.
Probability: 16.67%
1 in 6 chance
Probability distribution
View chart data as table
| Sum | Probability |
|---|---|
| 2 | 2.78% |
| 3 | 5.56% |
| 4 | 8.33% |
| 5 | 11.11% |
| 6 | 13.89% |
| 7 | 16.67% |
| 8 | 13.89% |
| 9 | 11.11% |
| 10 | 8.33% |
| 11 | 5.56% |
| 12 | 2.78% |
Full distribution table
| Sum | Ways | Probability | Cumulative (at least) |
|---|---|---|---|
| 2 | 1 | 2.78% | 100.00% |
| 3 | 2 | 5.56% | 97.22% |
| 4 | 3 | 8.33% | 91.67% |
| 5 | 4 | 11.11% | 83.33% |
| 6 | 5 | 13.89% | 72.22% |
| 7 | 6 | 16.67% | 58.33% |
| 8 | 5 | 13.89% | 41.67% |
| 9 | 4 | 11.11% | 27.78% |
| 10 | 3 | 8.33% | 16.67% |
| 11 | 2 | 5.56% | 8.33% |
| 12 | 1 | 2.78% | 2.78% |
Dice probability basics
With a single fair die, each face has an equal probability: 1/6 for a six-sided die, 1/20 for a twenty-sided die. When you roll multiple dice and add them together, the probabilities are no longer equal — middle values are far more likely than extremes. Two six-sided dice produce 7 most often (6 out of 36 possible combinations = 16.67%).
Why this matters for gambling
Dice games like craps are built on these probabilities. The pass line bet in craps has a 1.41% house edge — one of the best bets in the casino — because the rules are calibrated around the exact probability distribution of two dice. Understanding which sums are likely and which are rare is the foundation of playing any dice game intelligently.
Fair dice vs loaded dice
This calculator assumes perfectly fair dice where each face is equally likely. Physical dice can have slight manufacturing imperfections, but regulated casino dice are precision-made to tolerances of 1/10,000 of an inch. Online dice games use random number generators that produce mathematically perfect distributions.
How dice probability works
The formula
P(sum = k) = (1/SN) × Σ (-1)j × C(N,j) × C(k−S×j−1, N−1)
Where N is the number of dice, S is the number of sides per die, and k is the target sum. This calculator uses dynamic programming for efficiency — computing the full distribution in a single pass rather than evaluating each sum independently.
Why middle values dominate
When rolling multiple dice, there are many more combinations that produce middle sums than extreme ones. With two six-sided dice, there are 6 ways to roll a 7 but only 1 way to roll a 2 or 12. This is why craps proposition bets (targeting specific numbers) carry higher house edges — they're betting on rare outcomes.
Practical applications
Beyond casino games, dice probabilities are fundamental to tabletop RPGs (what are the odds of rolling 18 on 3d6?), board games, and probability education. Understanding these distributions helps you make informed decisions whenever dice are involved — whether that's a craps table or a D&D dungeon.