2 Dice Roller Calculator
Enter the type of dice and your target sum to see the exact probability of rolling that total, the odds against it, the expected (average) sum, and the full probability distribution from the lowest to the highest possible result. Works with D4, D6, D8, D10, D12 and D20. Results update as you type.
How the probability is calculated
When you roll two fair dice, every ordered pair of face values is equally likely. With two D6 dice there are 6 x 6 = 36 such pairs. To find the probability of a particular sum, count how many pairs add up to that total and divide by 36. For the sum of 7, the pairs (1,6), (2,5), (3,4), (4,3), (5,2) and (6,1) all work, giving 6/36 = 16.67%. The distribution is symmetric: sums near the middle of the range are more likely because they can be made more ways, while the extremes (2 and 12) can only be made one way each. This calculator uses the same combinatorial counting for any die type from D4 to D20.
Expected value and standard deviation
The expected (average) sum of two identical dice is always faces + 1. For D6 dice it is 7, for D8 dice it is 9, and so on. This follows because the expected value of one die is (faces + 1) / 2, and with two dice you simply double it. The standard deviation of the sum measures how spread out the results are: it equals the square root of 2 x (faces^2 - 1) / 12. For D6 this is about 2.42, meaning most rolls land within roughly two and a half pips of 7. Larger dice have wider distributions, making extreme sums relatively more reachable.
At-least and at-most probabilities
Exact-sum probability is only one of three useful queries. "At least X" (cumulative from above) answers questions like "what are the chances of rolling a combined 9 or higher in a board game?" It sums the probabilities for every sum from X up to the maximum. "At most X" works the same way from below. Because the distribution is symmetric, P(sum >= k) = P(sum <= 2*faces + 2 - k) for any die type. These cumulative probabilities are essential in games like Craps, Backgammon and Dungeons and Dragons, where thresholds matter more than exact values.
Different dice types: D4 to D20
While the six-sided die (D6) is the most familiar, tabletop games use a full family of polyhedral dice. The D4 has the smallest range and tightest distribution (sums 2 to 8), while the D20 spreads sums from 2 to 40. Switching to a higher die type shifts the expected sum upward and widens the distribution, making any specific target sum less probable. The calculator handles all six standard polyhedral types: D4, D6, D8, D10, D12 and D20, using the same combinatorial formula throughout.
Sum probabilities for two standard D6 dice
| Sum | Ways to roll | Fraction | Probability |
|---|---|---|---|
| 2 | 1 | 1/36 | 2.78% |
| 3 | 2 | 2/36 | 5.56% |
| 4 | 3 | 3/36 | 8.33% |
| 5 | 4 | 4/36 | 11.11% |
| 6 | 5 | 5/36 | 13.89% |
| 7 | 6 | 6/36 | 16.67% |
| 8 | 5 | 5/36 | 13.89% |
| 9 | 4 | 4/36 | 11.11% |
| 10 | 3 | 3/36 | 8.33% |
| 11 | 2 | 2/36 | 5.56% |
| 12 | 1 | 1/36 | 2.78% |
Each of the 36 outcomes is equally likely. The sum of 7 is most probable.
Frequently asked questions
What is the most common sum when rolling two D6 dice?
A sum of 7 is the most likely result with two standard six-sided dice. It can be formed in six ways: (1,6), (2,5), (3,4), (4,3), (5,2) and (6,1), giving a probability of 6/36 or about 16.67%. The sums of 2 and 12 are the least likely, each achievable in only one way (1/36, about 2.78%).
How do I calculate the probability of rolling a specific sum?
Count how many ordered pairs of die faces add up to your target sum, then divide by the total number of outcomes (faces x faces). For two D6 dice the total is 36. For a sum of 9: the pairs are (3,6), (4,5), (5,4) and (6,3), so the probability is 4/36, about 11.11%.
What is the expected sum of two dice?
The expected sum equals the number of faces plus one. For two D6 dice it is 7. For two D8 dice it is 9. For two D20 dice it is 21. This is because the average face on one die is (faces + 1) / 2, and you are rolling two.
Does it matter if I roll both dice at once or one at a time?
No. Rolling two dice simultaneously and recording both outcomes has exactly the same probability distribution as rolling them sequentially. Every ordered pair (die1, die2) is equally likely in both cases, so the statistics are identical.
What does "at least" probability mean for dice?
"At least X" probability is the chance that the sum of two dice is X or higher. It equals the sum of the individual probabilities for every possible sum from X up to the maximum. For two D6 dice, the probability of rolling at least 10 is (3 + 2 + 1) / 36 = 6/36, about 16.67%.