Dice Probability Explorer
Compare the exact outcomes of summing dice, keeping the highest die or keeping the lowest die. Add a modifier, test a threshold, and explore the complete probability distribution.
Define the dice experiment
Use 1–100 dice with 2–100 sides each, numbered from 1 through the side count. Modifier and threshold accept integers from −1,000,000 to 1,000,000. All dice have the same sides and every face is equally likely. The modifier is added once after summing or keeping one die.
All three modes are calculated together. Keep highest and keep lowest retain exactly one die from the pool. With two d20, these are commonly called advantage and disadvantage. This does not keep several dice or reroll ties.
Compare the same threshold: 7
Select a mode to inspect its full distribution. The headline probability is inclusive: P(X ≥ 7), after the modifier.
Total equally likely ordered rolls: 36
Ordered rolls distinguish which die shows which face. For two d6, (1, 6) and (6, 1) are different outcomes. Keep modes preserve that same sample space. Tiny nonzero probabilities are marked instead of displayed as zero.
Sum all dice distribution
Each bar is one possible outcome. Gold marks the inspected outcome’s bar. The vertical scale fits this chart; use the numbers to compare modes. The table always lists individual outcomes.
P(X = 7): 6/36
Reduced: 1/6
Probability: 16.666667%
P(X ≤ 7): 21/36
CDF: 58.333333%
| Outcome | Exact count | P(X = x) | Count ≤ x | P(X ≤ x) |
|---|---|---|---|---|
| 1 | 2.777778% | 1 | 2.777778% | |
| 2 | 5.555556% | 3 | 8.333333% | |
| 3 | 8.333333% | 6 | 16.666667% | |
| 4 | 11.111111% | 10 | 27.777778% | |
| 5 | 13.888889% | 15 | 41.666667% | |
| 6 | 16.666667% | 21 | 58.333333% | |
| 5 | 13.888889% | 26 | 72.222222% | |
| 4 | 11.111111% | 30 | 83.333333% | |
| 3 | 8.333333% | 33 | 91.666667% | |
| 2 | 5.555556% | 35 | 97.222222% | |
| 1 | 2.777778% | 36 | 100% |
Exact mean: 7/1. Every row count uses denominator 36. Percentages round to six decimal places; exports retain full integer fractions.
Export the complete experiment
CSV includes every outcome for all three modes, regardless of the visible page. TXT includes exact threshold counts and means. Nothing is automatically saved.
Exact outcome counts for fair, independent, equally sided dice. Decimal percentages and means are rounded. No loaded-die, correlated-roll or gambling recommendations. Inputs stay in your browser. Calculations run in a cancellable worker with a 15-second timeout; incomplete results are never shown as exact.
Common questions
- What is the probability of rolling a total of seven on two d6?
- Six ordered rolls sum to seven: (1,6), (2,5), (3,4), (4,3), (5,2) and (6,1). There are 36 equally likely ordered outcomes, so the probability is 6/36, or 1/6, approximately 16.666667%.
- How are advantage and disadvantage calculated?
- Keep highest retains one highest die; keep lowest retains one lowest die. With two d20 these are commonly called advantage and disadvantage. The highest die is 20 in 39 of 400 ordered outcomes, or 9.75%. Ties need no reroll. For n fair s-sided dice, the highest value k has k^n − (k−1)^n outcomes.
- When is the modifier applied?
- The integer modifier is applied once, after the dice are summed or one die is kept. For example, two d6 summed with modifier −2 range from zero to ten. In keep-highest mode the same modifier gives a range from −1 to four.
- What do PMF, CDF and the threshold comparisons mean?
- The PMF gives P(X = x), the probability of one exact outcome. The CDF gives P(X ≤ x), the probability at or below it. The comparison cards also show P(X ≥ threshold), which includes equality. All values refer to the result after the modifier.
- Are the probabilities exact?
- Integer outcome counts are exact, calculated with BigInt convolution for sums and integer powers for highest/lowest. Fractions preserve that precision. Displayed percentages and means are rounded to six decimal places; tiny nonzero probabilities are marked explicitly. This assumes independent fair dice with equal sides.
- What are the calculation and chart limits?
- Use one to 100 dice, two to 100 sides per die, and integer modifiers and thresholds between −1,000,000 and 1,000,000. Calculations run in a cancellable worker and stop after 15 seconds without partial results. Charts group larger distributions into at most 60 adjacent intervals; the paginated table and CSV retain every individual outcome.
- Can I export all results?
- CSV includes all three distributions and every outcome, including exact counts, denominator, probability fractions and cumulative fractions. The visible page does not limit the export. TXT and copy include exact threshold comparisons and mean fractions. Inputs stay in your browser and are not automatically saved.
Exact outcome counts for fair, independent, equally sided dice. Decimal percentages and means are rounded. No loaded-die, correlated-roll or gambling recommendations.