Dice Probability Calculator
Calculate the probability of rolling specific sums with any number of dice. Find exact, at least, at most, or range probabilities for standard or custom dice. See also our Probability Calculator, Combinations Calculator, and Expected Value Calculator.
How to Use the Dice Probability Calculator
This calculator computes the probability of rolling specific sums with any combination of dice. It uses dynamic programming to count the exact number of ways to achieve each sum, giving precise probabilities rather than approximations. It works for standard 6-sided dice, 20-sided dice (d20), or any custom number of sides.
Enter the number of dice, sides per die, and your target sum. Choose whether you want the probability of rolling exactly that sum, at least that sum, at most that sum, or within a range. The calculator handles up to 10 dice efficiently. Results include the exact fraction, decimal probability, percentage, and odds notation.
The calculator also shows the expected (average) sum and standard deviation for your dice configuration. The expected sum of n dice with s sides each is n×(s+1)/2. For 2d6, the expected sum is 7. The distribution of sums approaches a normal (bell) curve as the number of dice increases, by the Central Limit Theorem.
Formula
Total Outcomes:
Total = sides^(number of dice)
Probability:
P(sum = k) = (ways to get sum k) / total outcomes
Expected Sum:
E(sum) = n × (s + 1) / 2
Variance of Sum:
Var(sum) = n × (s² - 1) / 12
Ways to get sum k (inclusion-exclusion):
W(k,n,s) = Σ (-1)^j × C(n,j) × C(k-j×s-1, n-1)
Example Calculation
Rolling 2d6, what is P(sum = 7)?
Total outcomes: 6² = 36
Ways to get 7: (1,6)(2,5)(3,4)(4,3)(5,2)(6,1) = 6 ways
P(sum = 7) = 6/36 = 1/6 ≈ 16.67%
Rolling 2d6, what is P(sum ≥ 10)?
Sum=10: 3 ways, Sum=11: 2 ways, Sum=12: 1 way
P(sum ≥ 10) = 6/36 = 1/6 ≈ 16.67%
Expected sum: 2×(6+1)/2 = 7
Reference Table
| Roll | Fraction | Percentage | Odds |
|---|---|---|---|
| 2d6 sum = 7 | 6/36 | 16.67% | 1 in 6 |
| 2d6 sum = 2 | 1/36 | 2.78% | 1 in 36 |
| 2d6 sum = 12 | 1/36 | 2.78% | 1 in 36 |
| 2d6 sum ≥ 10 | 6/36 | 16.67% | 1 in 6 |
| 1d20 = 20 | 1/20 | 5.00% | 1 in 20 |
| 3d6 sum = 10 | 27/216 | 12.50% | 1 in 8 |
| 3d6 sum ≥ 16 | 6/216 | 2.78% | 1 in 36 |
| 1d6 = 6 | 1/6 | 16.67% | 1 in 6 |
Frequently Asked Questions
Why is 7 the most common sum for 2d6?
Seven has the most combinations: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) — six ways out of 36 total. Each sum from 2 to 12 has a different number of combinations, forming a triangular distribution. Sums near the middle (7) have the most ways to occur, while extremes (2 and 12) have only one way each. This is why 7 is the most common roll in games like Craps.
How does the number of dice affect the distribution?
With one die, all outcomes are equally likely (uniform distribution). As you add more dice, the sum distribution becomes bell-shaped (approximately normal) by the Central Limit Theorem. With 2d6, the distribution is triangular. With 3d6 or more, it closely resembles a normal curve centered at n×(s+1)/2. More dice means the distribution is more concentrated around the mean, with extreme values becoming increasingly unlikely.
What dice are used in tabletop RPGs?
Common RPG dice include: d4 (tetrahedron), d6 (cube), d8 (octahedron), d10 (pentagonal trapezohedron), d12 (dodecahedron), and d20 (icosahedron). The d20 is most iconic, used for attack rolls and skill checks in D&D. This calculator works with any number of sides. For a d20, each face has exactly 5% probability. Rolling multiple dice (like 2d6 for damage) creates non-uniform distributions.
Are dice rolls truly random?
Physical dice are approximately random but not perfectly so. Factors like weight distribution, surface texture, throwing technique, and landing surface can introduce slight biases. Casino dice are precision-manufactured to minimize bias. For mathematical purposes, we assume ideal fair dice where each face has exactly equal probability. Digital random number generators can provide better uniformity than physical dice for applications requiring true randomness.
How do I calculate probability for non-standard dice?
This calculator works with any number of sides. Enter the number of sides per die (e.g., 8 for d8, 20 for d20, 100 for d100). The formulas are the same regardless of the number of sides — only the total outcomes (sides^dice) and the distribution shape change. For dice with non-standard numbering (like a d10 numbered 0-9), adjust your target sum accordingly.
What is the probability of rolling all sixes?
For n standard dice, the probability of all sixes is (1/6)^n. For 2 dice: 1/36 ≈ 2.78%. For 3 dice: 1/216 ≈ 0.46%. For 5 dice (Yahtzee): 1/7776 ≈ 0.013%. The probability decreases exponentially with each additional die. More generally, for s-sided dice, the probability of all showing the same specific face is (1/s)^n. The probability of any matching set (all same, any face) is (1/s)^(n-1).