Probability
Dice Distribution Calculator
Calculate outcome count, possible range, target ways and probability, cumulative probability, expected total, and standard deviation. The custom probability histogram shows the exact shape rather than flattening the results into unrelated metrics.
Total equally likely outcomes-
Minimum possible total-
Maximum possible total-
Ways to roll the target-
Probability of exact target-
Probability of total at or below target-
Expected total-
Standard deviation of total-
Decision view
Exact dice-sum probability distribution
Exact dice-sum probability distributionEvery possible total is plotted and the entered target is highlighted.
Exact scenario comparisonTarget total changes while all other entered assumptions remain constant.
| Target total | Total equally likely outcomes | Minimum possible total | Maximum possible total | Ways to roll the target | Probability of exact target | Probability of total at or below target | Expected total | Standard deviation of total |
|---|
Probability detail
Complete total distribution and target summary
How to use Dice Distribution Calculator
- Enter identical fair dice and a target total inside the possible range.
- Read the highlighted bar for exact probability and cumulative output for at-most probability.
- Model rerolls, modifiers, kept dice, advantage, or nonstandard faces with their actual rules instead of this base distribution.
Calculator guide
Understanding Dice Distribution Calculator
Sums of fair dice are not equally likely. This calculator enumerates the discrete distribution, highlights the selected total, and separates outcomes below, equal to, and above the target.
Calculation method
How the calculation works
Enumerate the exact sum distribution for independent fair dice with identical side counts, then report exact-target and cumulative probabilities together with the theoretical mean and standard deviation. Convolve the uniform one-die distribution for the entered dice count, count ordered outcomes for every sum, and divide each count by sides raised to dice count.
Game interpretation
Match the probability to the actual rule
Most mistakes come from answering a nearby but different probability question.
Worked situations
Practical examples
- Two six-sided dice have 36 ordered outcomes.
- Six outcomes total seven, so the exact probability is 6/36.
- Totals near the mean are more likely than extreme totals.
Better inputs
Useful tips
- Use exact probability for a single required sum and cumulative probability for thresholds.
- Check whether a game treats dice as distinguishable or applies conditional rerolls.
- Use the complete table when payout changes by total.
Before relying on the result
Limitations and common mistakes
- Dice are assumed independent, fair, identical, and numbered consecutively from 1 to the entered side count.
- Physical bias, custom faces, exploding dice, dropped dice, modifiers, and conditional rules are excluded.
- Large combinations may be displayed with rounded percentages even when counts remain exact.
Reference
Key terms
- Ordered outcome
- One face result for each distinguishable die.
- Ways
- Number of ordered outcomes producing a particular sum.
- Exact probability
- Ways for the selected sum divided by all outcomes.
- Cumulative probability
- Probability of the selected sum or any lower sum.
Important note
Calculated directly from the entered values using the displayed formula and rounding settings.
Frequently asked questions
Why is seven more likely than two with two dice?
Seven has six ordered combinations while two has only one.
Does dice order matter?
It matters when counting equally likely ordered outcomes, even if only the sum is reported.
Can I use this for weighted dice?
No. Weighted faces require individual probabilities.
Why is the distribution symmetric?
For fair consecutive faces, sums pair around the mean with equal counts.