Calculator Inputs
Example Data Table
| Dice | Sides | Query | Ways | Probability |
|---|---|---|---|---|
| 2 | 6 | Exact sum 7 | 6 | 16.666667% |
| 3 | 6 | Range 10–12 | 108 | 50.000000% |
| 2 | 8 | Sum ≥ 10 | 21 | 32.812500% |
Formula Used
For n dice with sides s₁, s₂, …, sₙ, the total ordered outcomes are: T = s₁ × s₂ × … × sₙ.
The number of ways to reach each sum is computed with dynamic programming (convolution):
- Start with ways₀(0)=1.
- For each die, update: waysᵢ(sum) = Σ waysᵢ₋₁(sum − face) over all faces.
Probability for any condition is P = Ways ÷ T, shown as a percentage.
How to Use This Calculator
- Choose Standard dice or Custom dice list.
- Pick an analysis type (Exact, ≥, ≤, or Range).
- Enter the target sum or range values.
- Press Calculate to see the results above the form.
- Use Download CSV or Download PDF after calculating.
Insights and Practical Data
Outcome Space and Growth
Total outcomes grow multiplicatively: for 6-sided dice the count is 6n. With 2 dice you have 36 outcomes, 5 dice gives 7,776, and 10 dice reaches 60,466,176. This calculator keeps exact “ways” per sum, then converts to probabilities by dividing by the total outcomes.
Classic 2d6 Benchmark
For two 6-sided dice, sums range from 2 to 12. The peak occurs at 7 with 6 ways, so P(sum=7)=6/36=16.666667%. The edge sums 2 and 12 each have 1 way, so 2.777778% each. Use this benchmark to verify setups quickly.
Ranges and Threshold Decisions
“At least” and “at most” queries answer gameplay thresholds. For 2d6, P(sum≥10) = (3+2+1)/36 = 16.666667%. Range queries are helpful for skill checks; for 3d6, the middle region (10–12) is typically much denser than the tails, improving consistency.
Custom Dice Mixes
Mixed dice (like 6,6,8) change both the maximum sum and the shape. The total outcomes become 6×6×8=288. Because the 8-sided die adds broader spread, probabilities flatten compared with 3d6. This tool computes the exact distribution using convolution across each die’s faces.
Expected Value and Spread
The calculator reports an estimated expected sum and variance from the distribution. For identical dice, E = n×(s+1)/2. Example: 4d6 has E=14.0. Larger variance means outcomes swing more; narrower variance means your sums cluster near the center, which is visible in the Plotly bars.
Reading the Graph and Table Together
Use the table for exact “ways” and the Plotly chart to compare shapes at a glance. The highest bar indicates the most likely sum. Toggling the “ways” series can help when totals are huge, while probability (%) keeps comparisons consistent across different dice sizes and counts.
FAQs
1) What does “ways” mean?
“Ways” counts ordered outcomes. Rolling 2 then 5 is different from 5 then 2, so both are counted separately when calculating combinations and probabilities.
2) Why can probabilities show “—”?
Very large configurations can exceed floating-number limits for display. The exact integer counts remain correct, but the percentage may be unavailable when numbers overflow typical numeric ranges.
3) How is the distribution computed?
It uses dynamic programming (convolution). Each new die “adds” its face values to the prior distribution, accumulating counts for every possible sum.
4) Can I model mixed dice like 6,6,8?
Yes. Select Custom dice list and enter comma-separated sides. The tool multiplies those sides for total outcomes and builds the exact sum distribution.
5) What’s the difference between exact and range?
Exact targets one sum value. Range totals all sums between two bounds (inclusive), which is useful for “success windows” or banded scoring systems.
6) Do downloads include my latest calculation?
Yes. After you calculate, the CSV and PDF buttons export the most recent distribution, focus query, total outcomes, and the probability values shown on the page.