Dice Roll Probability Calculator
Compute the probability of rolling any target sum with any number of multi-sided dice. It counts the favorable combinations and divides by all equally likely outcomes, which is perfect for board-game and tabletop RPG design.
Last updated: May 2026
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About this calculator
The calculator finds P(sum = S) when you roll n dice that each have d sides. Every die is fair and independent, so there are d^n equally likely ordered outcomes in total. The probability is (number of ordered ways to reach the target sum S) ÷ d^n, where n is Number of Dice, d is Sides per Die, and S is Target Sum. Counting the favorable ways is the hard part: it equals the number of integer solutions to x₁ + x₂ + ... + xₙ = S with each xᵢ between 1 and d, which the tool computes by recursive enumeration. The distribution of the sum is triangular for two dice and approaches a bell curve as n grows, peaking at the mean S = n·(d+1)/2. Important edge cases: the sum is impossible (probability 0) when S < n (you cannot beat all-ones) or S > n·d (you cannot beat all-maximums). The result is symmetric about the mean, so S and (n·(d+1) − S) always share the same probability. Because outcomes are equally likely only for fair dice, weighted dice break the model.
How to use
Example 1 — two six-sided dice summing to 7. Enter Number of Dice = 2, Target Sum = 7, Sides per Die = 6. There are 6 ordered ways to make 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). Total outcomes = 6^2 = 36. Probability = 6/36 = 0.1667, or 16.67%. Verify: 7 is the mean of two dice, so it should be the single most likely sum — and it is. Example 2 — two six-sided dice summing to 4. Enter Number of Dice = 2, Target Sum = 4, Sides per Die = 6. The ordered ways are (1,3), (2,2), (3,1) → 3 favorable outcomes out of 36. Probability = 3/36 = 0.0833, or 8.33%. Verify by symmetry: a sum of 4 should be as likely as a sum of 10, and indeed (4,6),(5,5),(6,4) also gives 3/36 = 8.33%.
Frequently asked questions
Why is 7 the most likely sum on two dice?
Because 7 is the mean of two six-sided dice, n·(d+1)/2 = 2·7/2 = 7, and the distribution of a sum of dice is symmetric and peaks at its mean. There are six different ordered combinations that total 7, more than for any other sum, so it captures the largest share of the 36 equally likely outcomes. Sums farther from 7 have progressively fewer combinations, producing the familiar triangular shape from 2 up to 12. A common mistake is to count unordered pairs (treating (3,4) and (4,3) as one), which undercounts and gives wrong probabilities. Always count ordered outcomes so the denominator d^n stays consistent.
How do I find the probability of rolling a sum or higher instead of an exact sum?
This tool gives the probability of an EXACT target sum, so for 'S or higher' you must add the probabilities of S, S+1, all the way up to the maximum n·d. For example, on two dice the chance of exactly 10 is 3/36, but 10-or-higher is (3+2+1)/36 = 6/36 = 16.7%. Tabletop rules that say 'roll X or more to succeed' need this cumulative sum, not the single value. Forgetting to accumulate is a frequent error that makes success look rarer than it is. If you need many such thresholds, compute the full distribution once and take running totals.
What happens if the target sum is impossible?
The probability is exactly 0, and the calculator returns 0 for any sum below n or above n·d. With three six-sided dice, for instance, the smallest possible sum is 3 (all ones) and the largest is 18 (all sixes), so asking for 2 or 19 yields 0. This is a useful sanity check: if you expected a nonzero answer, you probably mismatched the number of dice or the number of sides. It also reflects a real design constraint — you cannot require players to roll a total their dice physically cannot produce. Always confirm your target lies in the achievable range n to n·d.
Does this assume the dice are fair, and what if they are not?
Yes. The formula divides favorable outcomes by d^n, which is only valid when every face of every die is equally likely and the dice are independent. Loaded or weighted dice violate this, so the true probabilities would differ and this calculator would mislead you. Casino-style or novelty dice with non-standard faces (for example a die labeled 1,1,2,3,3,4) also break the equal-likelihood assumption. For such cases you need a custom model that assigns each face its real probability and convolves the individual distributions. Use this tool only for standard, fair, independent dice.
When should I NOT use a dice-sum probability calculator?
Avoid it when the dice are not identical and fair, when you keep or reroll dice (as in many modern RPG mechanics), or when 'success' depends on individual dice rather than their total. Pool mechanics like 'count how many dice show 5 or 6' are binomial problems, not sum problems, and need a different model. Advantage/disadvantage systems that take the highest or lowest of several dice also fall outside this tool. Likewise, exploding dice (reroll and add on a maximum face) change the outcome space entirely. Match the calculator to whether your rule is about a total, a count, or an extreme value.