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Dice Roll Probability Calculator
What are the odds of rolling a total of 18 or higher with three 6-sided dice? What’s the probability of rolling a natural 20? Calculate the exact probability of any dice outcome — for any number of dice, any number of sides, and any target result.
From board games to tabletop RPGs to casino craps, dice probability is one of the most practical applications of combinatorics. The math is exact and the patterns are often surprising.
Dice Roll Probability
2d6 • target: 7
Probability: 0%
Roll Statistics
Common RPG Reference
| Roll | Probability | 1 in X |
|---|
Full Sum Distribution
| Sum | Ways | Probability | Cumul. (≤) | Cumul. (≥) |
|---|
"Rolling two dice and asking for a 7 gives you 6 ways out of 36 — a 16.7% chance. Rolling for a 2 gives you 1 way out of 36 — a 2.8% chance. The shape of the distribution is what separates intuition from mathematics."
— Dice Probability Fundamentals
How dice probability works
A single fair d6 has a uniform distribution: each face (1–6) has exactly 1/6 probability. When you roll multiple dice and sum them, the distribution stops being uniform and becomes bell-shaped — because there are many more ways to roll a middle value than an extreme value.
For 2d6, there is only 1 way to roll a 2 (1+1) and only 1 way to roll a 12 (6+6), but there are 6 ways to roll a 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1). This is why 7 is the most common result with two dice — and why craps is built around it. The total number of combinations is 6² = 36, and 7 occurs 6/36 ≈ 16.7% of the time.
The calculator uses polynomial multiplication (convolving the distributions of each die) to compute exact probabilities for any combination of dice. Adding more dice or dice with more sides shifts the bell curve and narrows the relative spread.
lightbulb Common Dice Probability Reference
| Roll | Prob. | 1 in X |
|---|---|---|
| 2d6 = 7 (craps point) | 16.67% | 6 |
| 2d6 = 12 (boxcars) | 2.78% | 36 |
| 1d20 = 20 (nat 20, D&D) | 5.00% | 20 |
| 1d20 = 1 (nat 1, crit fail) | 5.00% | 20 |
| 4d6 drop lowest ≥15 (D&D ability) | 9.10% | 11 |
| 3d6 = 18 (max stat roll) | 0.46% | 216 |
| 2d6 ≥ 10 (succeed at task) | 16.67% | 6 |
| 1d6 = 6 five times in a row | 0.013% | 7,776 |
Dice Probability FAQs
Why is 7 the most common sum with two d6?
Because there are more combinations that produce 7 than any other sum. There are 6 ways to make 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1) out of 36 total combinations. The next most common are 6 and 8, each with 5 ways. The pattern is a triangle: the probability increases toward the middle and decreases toward the extremes, because middle sums have more paths to reach them.
What does “4d6 drop lowest” mean in D&D?
Roll four six-sided dice, discard the lowest result, and sum the remaining three. This method generates character ability scores in Dungeons & Dragons and similar games. The expected value is about 12.24 (vs. 10.5 for a straight 3d6 roll), and it reduces the chance of very low scores while keeping some variance. It makes characters statistically better than average without guaranteeing high rolls.
How do craps odds work with dice?
Craps is built on 2d6 probability. On the come-out roll, a 7 or 11 wins (8/36 = 22.2%), 2, 3, or 12 loses (4/36 = 11.1%), and any other number sets the “point.” Once a point is set, you must roll that number again before rolling a 7. Since 7 is the most common sum, the house has a built-in advantage. The pass line house edge of 1.41% emerges from these probabilities.
Terminology
Dice Notation (NdX)
Standard notation for dice rolls. N = number of dice, X = number of sides. 2d6 = two six-sided dice. 1d20 = one twenty-sided die. 3d8+5 = three eight-sided dice plus 5. Common in tabletop RPGs and game design.
Probability Distribution
The complete mapping of every possible outcome to its probability. For 2d6, the distribution runs from sum=2 to sum=12, with probabilities 1/36, 2/36, ..., 6/36, ..., 1/36. The distribution visualizes which outcomes are likely vs. rare.
Convolution
The mathematical operation for combining the distributions of multiple independent dice. The probability distribution of the sum of two dice is the convolution of their individual distributions. This is what the calculator computes — multiplying polynomials whose coefficients represent each face’s probability.
Expected Value (Mean)
The average sum over many rolls. For a single dN, the expected value is (N+1)/2. For a d6: (6+1)/2 = 3.5. For 2d6: 2×3.5 = 7. The expected value is always the midpoint of the range for a fair die.
Advantage / Disadvantage (D&D 5e)
Roll two d20s and take the higher (advantage) or lower (disadvantage) result. With advantage, the probability of rolling 20 doubles from 5% to 9.75%. With disadvantage, the probability of rolling 1 also nearly doubles. The calculator’s keep-highest option models advantage.
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