Estimate probabilities for exact sums, ranges, repeated faces, and special multi-die events. The calculator also shows distribution insights, exportable outputs, and a practical chart.
Calculator Inputs
Distribution Graph
The chart displays the full sum distribution for the selected dice and side count, helping you inspect skew, spread, and peak outcomes.
Example Data Table
| Scenario | Dice | Sides | Event | Probability |
|---|---|---|---|---|
| Classic trio | 3 | 6 | Exact sum = 10 | 27/216 = 0.1250 |
| Higher threshold | 4 | 6 | At least sum 18 | 206/1296 = 0.15895 |
| Face repetition | 5 | 6 | At least one 6 | 1 - (5/6)^5 = 0.59812 |
| Uniform match | 4 | 8 | All dice the same | 1/512 = 0.00195 |
Formula Used
General probability rule: P(E) = favorable outcomes ÷ total outcomes.
Total outcomes for n dice with s sides: sn.
Exact sum and range queries: build the exact count of ways for each sum, then add the counts matching the requested event.
At least one chosen face: 1 − ((s−1)/s)n.
Exactly k copies of one face: C(n,k) × (1/s)k × ((s−1)/s)n−k.
All same: 1/sn−1. All distinct: s × (s−1) × ... × (s−n+1) ÷ sn.
How to Use This Calculator
- Enter how many dice you will roll.
- Set the number of sides on each die.
- Choose the event type you want to evaluate.
- Fill the needed target sum, range, face value, or k count.
- Press the calculate button to show the result section above the form.
- Review the probability, percentage, fraction, and distribution graph.
- Export the generated report as CSV or PDF when needed.
Frequently Asked Questions
1. What does this calculator measure?
It measures the probability of several multi-dice events, including exact sums, cumulative sums, repeated faces, at least one chosen face, and special patterns like all same or all distinct.
2. Why do exact sums need a distribution table?
Different sums can occur in different numbers of ways. A full distribution captures every reachable total and counts how often each total appears across all possible outcomes.
3. Can I use nonstandard dice?
Yes. You can change the side count, so the tool works for d4, d6, d8, d10, d12, d20, and other uniform dice with equal face probabilities.
4. What does “at least one chosen face” mean?
It means one or more dice show the selected face value. The formula uses the complement rule, which subtracts the probability of getting no such face from one.
5. When is “all distinct” impossible?
It becomes impossible when the number of dice exceeds the number of sides. You cannot assign unique faces to every die if there are fewer available face values.
6. Does the calculator assume fair dice?
Yes. Every formula here assumes each face is equally likely and every die is independent. Loaded or biased dice require a different probability model.
7. What do variance and standard deviation show?
They describe how spread out the sum distribution is. Larger values indicate more variability in totals, while smaller values indicate outcomes cluster more tightly around the expected sum.
8. Why export CSV or PDF reports?
Exports make it easier to document game analysis, compare class examples, share calculations, or save a clean summary of the selected dice setup and resulting probabilities.