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Enter your values, then click Calculate result.How this calculator helps
Compute exact outcome counts and sum probabilities for fair independent numbered dice, with exactly, at-most and at-least comparisons. Enter a whole-number dice count from one through twenty and a face count from two through one hundred. Each die is assumed to have one equally likely face for every integer from one through the chosen side count. All dice share that side count and are independent. Loaded dice, repeated labels, unequal dice and special game mechanics require different distributions and are outside this worksheet.
How to use it
- 1
Select the correct input roles for dice sum probability calculator and enter the values described below. The demonstration defaults illustrate the method; they are not independently verified personal measurements or live market data.
- 2
Review ordered outcome space and the original source record. Match units, signs and the chosen mode before submitting, rather than relying on a familiar-looking default number.
- 3
Click Calculate result to submit the current fields. Editing an input preserves the prior submitted output until you calculate again; the pending-change message distinguishes that saved output from the new values.
- 4
Check the labeled output against the worked example and independent verification steps. Review what this page does before using the result in another document, and keep the complete input basis with a copied answer.
Formula and methodology
Total ordered outcomes = sⁿ. Count sum outcomes by convolution of the uniform faces 1…s. Probability = favorable count/sⁿ.
The calculator applies the displayed arithmetic to the values entered on this device. It does not silently load a local tax rate, currency conversion or commercial assumption.
Worked calculation example
Two six-sided dice have 36 ordered outcomes. Exactly six of them sum to seven: (1,6), (2,5), (3,4), (4,3), (5,2) and (6,1). Probability is 6/36=1/6, about 16.666666667%. At most seven includes 21 outcomes, or 58.333333333%. The expected sum is seven, but that does not make seven a guaranteed next roll.
How to interpret your result
Exactly counts only the target sum. At most includes the target and every lower possible sum; at least includes the target and every higher possible sum. The target may be zero through two thousand. A target outside the attainable interval gives zero or one as appropriate instead of an arbitrary error. The attainable sums range from the dice count to dice count multiplied by sides.
For different inputs or formulas, use Probability Calculator; Permutation and Combination Calculator; Binomial Distribution Calculator — Exact and Tail Probabilities.
Related questions this calculator covers
- dice calculator
Scenario comparison
| Scenario | What it shows |
|---|---|
| One die | each attainable face has probability 1/s. |
| Minimum sum | all dice must show one, giving one favorable ordered outcome. |
| Complement | P(sum≤t)+P(sum≥t+1)=1. |
Common mistakes to avoid
- Assuming each sum is equally likely.
- Excluding the target from an inclusive comparison.
- Using a fair-dice model for reroll or drop-lowest rules.
For two six-sided dice, enumerate all thirty-six ordered pairs and count the target sums. Confirm that the full count sums to sides raised to dice count. Check symmetry around the expected sum and verify the inclusive complementary probabilities at neighboring thresholds. These checks validate the stated model rather than proving a physical die is fair.
Authoritative reference and additional source reference. Method and scope reviewed on 5 October 2026. SolvePilot provides the original worked example and bounded browser implementation. Editorial and arithmetic review by Mohammad Qasim does not certify user measurements, a real contract or an individual professional decision. The reference supplies method, unit or source-record context; it does not approve this implementation or its inputs.What can affect the result?
Fair numbered dice
Enter a whole-number dice count from one through twenty and a face count from two through one hundred. Each die is assumed to have one equally likely face for every integer from one through the chosen side count. All dice share that side count and are independent. Loaded dice, repeated labels, unequal dice and special game mechanics require different distributions and are outside this worksheet.
Ordered outcome space
For two dice, rolling one then six differs from six then one, even though their sums agree. Counting these ordered outcomes makes every elementary result equally likely under the model. A sum near the middle has more contributing outcomes than an extreme sum. The calculator builds exact sum counts with repeated integer convolution rather than simulating a random sample and treating sampled frequency as theoretical probability.
Choose the comparison
Exactly counts only the target sum. At most includes the target and every lower possible sum; at least includes the target and every higher possible sum. The target may be zero through two thousand. A target outside the attainable interval gives zero or one as appropriate instead of an arbitrary error. The attainable sums range from the dice count to dice count multiplied by sides.
Counts and rounded percentage
Favorable outcomes and all outcomes are exact integers, and the reduced probability fraction preserves exact meaning. The displayed percentage converts their ratio to floating point and rounds the decimal presentation. Very small nonzero probabilities use scientific notation. The expected sum is dice count times the average face value; it describes a long-run mean, not the most recent roll or an assured outcome.
What this page does
This deterministic probability worksheet does not roll physical dice, produce secure randomness, choose a game strategy or guarantee a gambling result. It does not account for rerolls, advantage, dropped dice, exploding dice or conditional bonuses. Changing the assumed side count changes the entire distribution. Save the dice count, sides, target and comparison together so an exact fraction remains attached to the model that produced it.
Privacy and browser processing
Values entered on this page are processed in the current browser session. SolvePilot does not require an account and does not receive the values entered into the calculator. Refreshing or closing the page clears the working values unless the browser itself restores a previous session. Avoid entering identifying or account information because the calculation needs summary values only.
Accuracy and verification
Accuracy depends first on input quality. Confirm definitions, scales, dates and source information before entering a value. Keep an independent record of any result used for planning because this page does not create an official statement or retain a calculation history.
Limits of this estimate
This deterministic probability worksheet does not roll physical dice, produce secure randomness, choose a game strategy or guarantee a gambling result. It does not account for rerolls, advantage, dropped dice, exploding dice or conditional bonuses. Changing the assumed side count changes the entire distribution. Save the dice count, sides, target and comparison together so an exact fraction remains attached to the model that produced it.
Sources and review information
Frequently asked questions
Does this roll dice?＋
No. It counts outcomes for an explicit probability model and returns the same result for the same inputs.
Why are central sums more likely?＋
More ordered face combinations produce them. Each combination is equally likely, but each distinct sum is not.
Does at least include the target?＋
Yes. Both cumulative comparisons include their endpoint: at least means greater than or equal to the target.
Can dice have different side counts?＋
Not in this page. All dice use the same numbered fair-face model.
Is the percentage exact?＋
The counts and reduced fraction are exact. The displayed percentage is a rounded floating-point representation.