Dice Probability Calculator
Rolling exactly a 7 with two six-sided dice happens 16.67% of the time, more often than any other sum on 2d6. This dice probability calculator counts every way a target total can appear across a pool of standard dice and converts that count into a percentage. Enter how many dice you are rolling, how many sides each die has, the sum you care about, and whether you want that sum exactly, that sum or higher, or that sum or lower. The tool then builds the full outcome distribution for that pool and reports the exact odds, the raw ways-to-total-outcomes ratio, and the smallest and largest sums the pool can ever produce. This is a different question from asking what a dice pool averages on a typical roll. A pool can average 7 while still landing on 7 itself less than one time in five, because many other sums also share the middle of the range. Knowing the average tells you where results center. Knowing the probability of one specific sum, or a band of sums, tells you how often that particular outcome actually shows up. That distinction matters for tabletop hit-chance math, board game strategy, dice-pool game design, and probability coursework where the question is phrased as a chance, not an expectation.
Quick answer
Every face on a fair die from 1 to S is equally likely, so every combination of faces across N dice is equally likely too.
Probability
16.67%
Ways / total outcomes
6/36
Minimum sum
2
Maximum sum
12
What this tells you
- •Every face on a fair die from 1 to S is equally likely, so every combination of faces across N dice is equally likely too.
- •The total number of equally likely outcomes for N dice with S sides each is S raised to the power of N.
- •Some sums have more ways to occur than others because more face combinations add up to them. That is why 7 beats every other 2d6 sum.
- •"Exactly" asks for one specific sum. "At least" adds up every sum from your target through the highest possible roll. "At most" adds up every sum from the lowest possible roll through your target.
- •A target sum below the pool's minimum or above its maximum still returns a valid answer of 0% or 100% for the comparison modes, since the pool always or never satisfies it.
- •The distribution gets more bell-shaped as you add dice, so middle sums pull further ahead of the extremes the more dice you roll.
How to Use
- 11. Enter the number of dice in the pool. For 2d6, type 2. For 3d6, type 3.
- 22. Enter the number of sides on each die. Use 6 for standard dice, 20 for a d20, or any value up to 100.
- 33. Enter the target sum you want to test, such as 7 for a lucky roll or 10 for a damage threshold.
- 44. Choose a mode. Pick Exactly for one specific total, At Least for that total or higher, or At Most for that total or lower.
- 55. Read the probability as a percentage first. It is the long-run share of rolls that satisfy your condition.
- 66. Check the ways and total outcomes figures underneath if you want the raw counting fraction behind the percentage, plus the pool's minimum and maximum possible sums for context.
How It Works
Formula
P = ways(condition) / S^NIn this formula, N is the number of dice, S is the number of sides on each die, and ways(condition) is how many of the S^N equally likely face combinations satisfy the chosen condition. The calculator finds ways(condition) by building the full sum distribution through convolution: it starts with a single die's distribution (1 way for each face 1 through S), then folds in one additional die at a time, adding every face value to every partial sum already tracked. After N dice, every possible total from N up to N x S has a ways count attached. For "exactly T", ways is just the count at T. For "at least T", ways is the sum of counts from T up to the maximum. For "at most T", ways is the sum of counts from the minimum up to T. Dividing that count by S^N gives the probability. Example: with 2d6, the total 7 has 6 winning face combinations (1-6, 2-5, 3-4, 4-3, 5-2, 6-1) out of 36 possible rolls, so P = 6 / 36 = 16.67%.
Calculation note: values are processed in the order shown above, using the current input units.
Worked Examples
2d6 exactly 7
Six face combinations add up to 7 on two six-sided dice: 1-6, 2-5, 3-4, 4-3, 5-2, and 6-1. No other sum on 2d6 has that many combinations, which is why 7 is the single most common roll in games built on two six-sided dice.
2d6 at least 10
Rolling 10 or higher on 2d6 means landing on 10, 11, or 12. Those three sums have 3, 2, and 1 winning combinations, for 6 total out of 36 possible rolls. It is a coincidence of this particular pool that the answer matches the exactly-7 case above.
1d20 at most 5
A single d20 has 20 equally likely faces, and 5 of them (1, 2, 3, 4, and 5) satisfy an at-most-5 condition. With only one die, the probability calculation collapses to simple counting: 5 qualifying faces out of 20 total faces is 25%.
3d6 exactly 10
Adding a third die spreads 216 total outcomes across sums from 3 to 18, and 27 of those outcomes land on exactly 10. That makes 10 the peak of the 3d6 curve, tied with 11 for the most likely total, which is why tabletop designers often build encounter tables around that pair.
1d6 at least 4
Three of the six faces on a single d6, namely 4, 5, and 6, meet an at-least-4 condition, so the probability is exactly 50%. This is a useful sanity check case because it matches the intuitive coin-flip odds of rolling in the top half of a d6.
2d6 Sum Probability Table
Every possible total on two standard six-sided dice, with the number of face combinations that produce it and the resulting probability. Use this table to see why 7 dominates and why the tails at 2 and 12 are rare.
| Sum | Ways | Probability |
|---|---|---|
| 2 | 1 | 2.78% |
| 3 | 2 | 5.56% |
| 4 | 3 | 8.33% |
| 5 | 4 | 11.11% |
| 6 | 5 | 13.89% |
| 7 | 6 | 16.67% |
| 8 | 5 | 13.89% |
| 9 | 4 | 11.11% |
| 10 | 3 | 8.33% |
| 11 | 2 | 5.56% |
| 12 | 1 | 2.78% |
Why some sums are more likely than others
A single die is flat. Every face from 1 to S has the same 1-in-S chance, so a d20 gives no face a special edge over any other. The moment you add a second die, that flatness disappears. Each total from N up to N x S is built from a different number of face combinations, and totals near the middle of the range have far more combinations than totals at the extremes.
Take 2d6 as the clearest case. Only one combination makes 2 (a 1 and a 1) and only one makes 12 (a 6 and a 6), but six different combinations make 7. As you add more dice, this pattern gets stronger, not weaker. The distribution for 3d6 already looks noticeably more peaked in the middle than 2d6, and by the time you reach a pool of 6 or more dice, extreme sums become vanishingly rare compared to sums near the center.
This is the same convolution effect behind the central limit theorem, applied to a small, discrete case you can count by hand. It is also why board games and tabletop systems that use 2d6 or 3d6 for core mechanics tend to feel more predictable than a single d20, even though both pools can technically reach a wide range of totals.
Common mistakes
- Assuming every sum on a multi-die pool is equally likely. Only a single die has that property. Add a second die and the middle sums pull ahead of the extremes.
- Confusing the average roll with the most probable roll. On 2d6 they happen to coincide at 7, but on many pools with modifiers or unusual sizes they do not line up at all.
- Forgetting that "at least" and "at most" include the target sum itself. At-least-10 on 2d6 counts 10, 11, and 12, not just 11 and 12.
- Mixing up total outcomes with total sums. Two six-sided dice have 36 total outcomes (6 x 6), not 11 total outcomes, even though there are only 11 possible sums from 2 to 12.
- Applying this calculator to dice with rerolls, exploding faces, or keep-highest and drop-lowest rules. Those mechanics change which combinations count and need different math than a plain sum distribution.
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