This dice probability calculator computes exact odds rather than simulating rolls. Give it a number of dice, a number of sides, an optional modifier and a target, and it returns the precise probability that the total meets your condition — along with the number of favourable outcomes, the total outcome space, the mean and the standard deviation, and a distribution strip showing how the sums around your target compare.
Arb Digital publishes it as part of a free tools library that includes a lot of statistics and probability work. Two neighbouring pages on the site cover related ground and it is worth being clear about the split. The dice roller generates actual random rolls, which is what you want when you need an outcome rather than an odds figure. The probability calculator handles general single- and two-event probability with unions, intersections and complements. This page is the only one that computes the full sum distribution for a pool of identical dice, which is a different calculation from either.
What This Dice Probability Calculator Does
The calculator builds the complete distribution of possible totals for your dice pool and then reads the answer off it. That means the headline probability is exact, not an estimate from a sample, and the outcome counts it reports are the real combinatorial counts.
Five conditions are supported: exactly the target, at least, at most, strictly greater and strictly less. The distinction between "at least 10" and "more than 10" matters far more often than people expect, because with 2d6 those two conditions differ by a full 8.33 percentage points. The modifier field handles rolls written in the familiar dice notation, so 3d6+2 is three dice, six sides and a modifier of two, and the target you enter is the finished total including that modifier.
The supporting figures cover the shape of the distribution rather than just the single answer. Mean total tells you the centre, standard deviation tells you how tightly the results cluster around it, and the distribution strip shows the probability of each individual total in a window around your target so you can see how quickly the odds fall away on either side.
How to Use It
- Enter the pool. Number of dice first, then sides. A standard pair of dice is 2 and 6; a single twenty-sided die is 1 and 20.
- Add any flat modifier. This is the fixed number added after the dice are summed, not a bonus applied to each die.
- Pick the condition. Choose carefully between "at least" and "more than" — they are not the same, and the difference is the probability of exactly hitting the target.
- Enter the target total. Include the modifier. If you want to know your chance of beating a difficulty of 15 with 1d20+4, the target is 15 and the modifier is 4.
- Read the distribution strip. The bars show individual totals near your target, which is how you spot whether a small change in the target moves the odds a lot or barely at all.
The Formula / How It's Calculated
With one die of s sides every face is equally likely, so each total has probability 1 divided by s. Adding a second die is a convolution: the probability of a total t is the sum, over every face value the new die could show, of the probability that the previous dice produced whatever remainder is needed. The calculator repeats that step once per die, so a pool of n dice takes n passes and produces an exact distribution.
The total outcome space is s to the power of n. Two six-sided dice give 36 ordered outcomes; five give 7,776. Favourable outcomes are the probability multiplied back by that total, which for whole-number counts recovers the classic combinatorial answer.
Worked example. Two six-sided dice, target exactly 7. The ordered pairs that sum to 7 are 1+6, 2+5, 3+4, 4+3, 5+2 and 6+1 — six of them out of 36, so the probability is 6 ÷ 36, which is 16.6667 per cent, or 1 in 6. Change the condition to "at least 10" and the favourable totals become 10, 11 and 12 with 3, 2 and 1 ways respectively, giving 6 ÷ 36 again — the same 16.6667 per cent, by coincidence.
The mean and standard deviation come from closed formulas rather than the distribution. The mean total is n × (s + 1) ÷ 2 plus the modifier, which for 2d6 is 2 × 3.5 = 7. The standard deviation is √(n × (s² − 1) ÷ 12), which for 2d6 works out at √(2 × 35 ÷ 12) = 2.4152. Note that the modifier shifts the mean but leaves the standard deviation untouched, because adding a constant moves a distribution without changing its spread.
Why More Dice Means Less Randomness
A single die is flat: every face is equally likely and the outcome is maximally unpredictable within its range. Two dice already form a triangle, with 7 six times as likely as 2 on a standard pair. Three dice produce a curve that is visibly bell-shaped, and by five or six dice the distribution is close enough to a normal curve that the approximation is useful for rough work.
The practical consequence is that the standard deviation grows with the square root of the number of dice while the mean grows linearly. Double the dice and the average doubles, but the spread only rises by about 41 per cent. Relative to the total, larger pools are therefore more predictable, which is why a game that wants dramatic swings uses one big die and a game that wants competence to dominate uses several small ones.
This is the same central-limit behaviour that makes the normal distribution ubiquitous in statistics, and the normal distribution calculator will give a close approximation for large dice pools. For small pools it will not — with two or three dice the discrete triangular shape is genuinely different from a bell curve, and the exact calculation on this page is the right one to use.
At Least, More Than, and the Off-by-One That Costs You
Almost every mistaken dice probability comes from a boundary error rather than a maths error. "At least 15" includes 15; "more than 15" does not. On a single twenty-sided die those two conditions differ by 5 percentage points, which is the entire probability of the boundary value.
The error is easy to make in the other direction too. A game rule that says "succeed on 15 or higher" is an at-least condition with six winning faces out of twenty, giving 30 per cent — not the 25 per cent you get by carelessly counting 16 through 20. Whenever a rule is written in words, translate it into one of the five conditions on this page before you calculate anything.
Modifiers create a related trap. Rolling 1d20+4 against a target of 15 means the raw die needs to show 11 or more, which is ten faces out of twenty, or 50 per cent. Entering the target as 11 with no modifier gives the same answer, but entering it as 15 with no modifier does not — the modifier and the target have to describe the same scale.
Ordered Outcomes, Combinations and Why 36 Not 21
A frequent objection to the 1-in-6 answer for a total of 7 is that there are only 21 distinct unordered pairs on two dice, not 36. Both counts are correct; only one of them is useful. The 36 figure counts ordered outcomes, treating a red 3 with a blue 4 as different from a red 4 with a blue 3. The 21 figure counts unordered combinations, merging those two.
The reason probability uses the ordered count is that ordered outcomes are equally likely and unordered ones are not. The unordered pair {3,4} occurs twice as often as the pair {3,3}, because there are two ways to roll it and one way to roll the double. Divide by 21 and you are implicitly claiming a double is as likely as a mixed pair, which physical dice comprehensively disagree with. Wolfram MathWorld's reference page on dice and their sum distributions sets out the generating-function treatment of the same result. If you need to count arrangements in their own right, the permutation calculator and the binomial coefficient calculator handle that side.
When the Dice Are Not Independent or Not Fair
Every number on this page assumes each die is fair and each roll is independent of the others. Both assumptions hold well for ordinary dice on a table, and physical bias in mass-produced dice is small enough to be invisible over the number of rolls a person will ever make. Casino-grade dice are machined precisely because that tiny bias becomes measurable over millions of throws.
Independence is the assumption that breaks first in practice. Mechanics that let you reroll a die, keep the highest of several, or drop the lowest do not produce the distribution calculated here — they produce a different one that is generally shifted upward and narrower. This calculator sums every die in the pool; it does not model keep-highest, exploding dice or reroll rules, and using it for those will overstate the chance of low totals. The distribution family it does cover is the plain sum, which is what the overwhelming majority of dice questions are about. The expected value calculator and the standard deviation calculator are useful when you have built a custom outcome table by hand, and the conditional probability calculator covers rolls that depend on an earlier result. The US National Institute of Standards and Technology's gallery of statistical distributions is a good reference for the continuous distributions that dice sums approach as the pool grows.
Arb Digital's free tools library covers probability, statistics and campaign maths, and the team is happy to talk through anything the calculators cannot answer.
Browse Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Confusing "at least" with "more than" — the gap between them is exactly the probability of the boundary total, which on a d20 is 5 percentage points.
- Dividing by unordered combinations — 21 unordered pairs on two dice are not equally likely, so probabilities must be taken over the 36 ordered outcomes.
- Applying a modifier to the target and the dice at once — decide whether the modifier is in the roll or in the target, and use it once.
- Assuming a bell curve for two or three dice — small pools are triangular or lumpy, and the normal approximation only becomes reasonable at five or six dice.
- Using these odds for keep-highest or reroll mechanics — those change the distribution, and a plain sum will understate your chance of a high total.
Related Free Tools From Arb Digital
Roll actual dice with the dice roller, handle general event probability with the probability calculator, and convert between odds and percentages with the odds probability converter. The coin flip probability calculator covers two-outcome trials, the poker odds calculator and the lottery odds calculator handle drawing without replacement, and the binomial distribution calculator deals with repeated pass-or-fail trials. Everything else is in the free online tools hub.
Frequently Asked Questions
Six of the 36 ordered outcomes sum to 7, so the probability is 16.6667 per cent, or 1 in 6. That makes 7 the most likely total on a standard pair, six times more likely than 2 or 12.
A dice roller produces actual random results, which is what you want when you need an outcome. This page never rolls anything — it computes the exact probability of a total from the full distribution, so the answer is the same every time you load it.
Because ordered outcomes are equally likely and unordered combinations are not. The pair three and four can occur two ways while a double three occurs one way, so dividing by 21 would treat unequal events as equal.
Enter 3 dice, 6 sides and a modifier of 2. The target you type is the finished total including the modifier, so a target of 14 means the three dice need to sum to 12.
No. Adding a constant shifts the entire distribution up or down without stretching it, so the mean moves by the modifier and the standard deviation stays exactly the same.
Relatively less. The mean grows in proportion to the number of dice while the spread grows only with its square root, so large pools cluster tightly around their average and extreme totals become very rare.
No. Those mechanics produce a different distribution from a plain sum, generally shifted higher and narrower. This calculator sums every die in the pool, so it will overstate your chance of low totals under those rules.
Yes. Any number of sides from 2 to 100 is accepted and the distribution is built from scratch each time, so unusual dice are handled exactly the same way as a familiar six-sided one.
