THE RULE BEFORE THE NUMBER
Tell DiceClarity how a roll works, choose the result you need, and calculate its probability. Start with an example, then change one rule at a time. Every answer assumes fair dice and independent rolls unless a stated repeat rule connects them.
At least 2 successes includes 2, 3, 4, and more. Exactly 2 excludes every larger result. At most includes the target and all smaller results. Between includes both endpoints.
For five d10 dice succeeding on 8+, the chance of at least two successes is 47.178%; exactly two is 30.87%. The average is 1.5 successes. An average of 1.5 does not mean you usually get two, and it is not an outcome a single roll can produce.
Use Add dice together for a total such as 2d6. Use Count successes when each die meeting a threshold scores a success. A fixed modifier is applied after the dice are scored; in a pool, it adds or subtracts successes.
With 4d6, keeping the highest two gives a 52.160% chance of at least 10. Rolling two separate 2d6 totals and keeping the higher gives 30.556%. Choose Which dice count? for individual dice, or Repeat the entire roll for whole-result advantage and disadvantage.
To drop the lowest die from 4d6, keep the highest 3. To drop the highest, keep the lowest 3. Keep/drop compares raw face values before summing.
Reroll each 1 once replaces an initial 1 even if the next roll is also 1. Reroll 1s until not 1 keeps rolling that die until a different face appears. Reroll every failure once replaces each face below the success threshold, once.
Reroll the lowest raw face once always replaces one lowest die, even when its old result was already successful. Tied lowest dice are equivalent here. This fixed policy is not an optimal strategy calculator.
Retry once if the target is missed is different again: finish the entire roll, compare it with the chosen target, and replace the entire result once only if it fails. Changing the target changes this rule's distribution. It cannot be used for opposed rolls, inverse searches, or explosions.
Exploding dice retain the maximum face's score and add another roll of that die. A new maximum repeats the process. For a sum, a 6 on d6 adds six points; for a success pool, it adds its success score. Explosions operate on each die separately, never on a pool's combined maximum.
In a pool, the maximum can score 2 successes, while a 1 can score −1. Cancellation takes precedence over threshold scoring. Custom face values accepts a comma-separated list such as -1,0,0,1,1,2; duplicate values remain separate equally likely faces. It does not model loaded dice with arbitrary real-valued weights.
Ones with no successes calculates a joint event: at least the selected number of 1s AND no dice reaching the success threshold, on the same initial roll. It does not assume the two conditions are independent, and does not add game-specific glitch rules.
Use Compare rules to evaluate A and B against the same event. The difference is reported in percentage points. Changing a chance from 30% to 40% adds 10 percentage points. Switch the chart to At least to compare performance across targets; the CSV preserves individual outcomes.
Use Opposed rolls when higher score wins. A strict win, a tie, and a strict loss are shown separately for one round. You can award ties to A, award them to B, or reroll both sides until the tie breaks. Rerolling both is conditioned on a non-tie; if every result ties, the calculator rejects the non-terminating rule.
Compare like units. A total of ten points and ten successes may not mean the same thing in your game. Natural critical hits, automatic misses, opposed roll-under rules, and game-specific tie breakers are not silently added.
Find dice needed checks success pools from 1 to 100 dice against an at-least target. It reports the first pool reaching your desired chance, and the previous pool's chance. For d10 succeeding on 8+, at least two successes reach 75% with 9 dice; 8 dice give 74.470167%.
If none of the tested pools qualifies, the result says so. It does not claim the target is impossible with more than 100 dice. A truncated exploding distribution must meet the goal with its lower-bound probability to qualify.
Group check accepts each person's individual pass chance. To require everyone, set the target to the number of people; for at least half, use half rounded up. People must be independent—shared bonuses, shared rerolls, or correlated outcomes need a different model.
Repeated attempts uses the same chance for every independent attempt. Three tries at 30% give a 65.7% chance of at least one success, but only a 2.7% chance that all three succeed.
Multi-stage attempts multiplies the conditional pass chances for the stages of each attempt, then counts completed attempts. Enter a stage chance conditional on having reached that stage. For a save that stops an attack, enter the chance the save fails, not its success chance. Separate attempts must be independent and use identical stage chances. This is a generic chain, not a complete combat or damage simulator.
DiceClarity uses deterministic probability calculations, not random sampling. Ordinary sums use discrete convolution; independent successes use binomial or Poisson-binomial distributions; keeping dice uses ordered face counts; selective rerolls use bounded outcome enumeration. Ordinary answers are calculated with JavaScript floating-point arithmetic and rounded for display, not symbolic exact arithmetic.
For exploding dice, the distribution is infinite. The engine follows terminating sequences until the remaining single-die tail is below 10−13. It retains the omitted probability mass and propagates that bound through independent combinations. It never silently renormalizes a truncated distribution. Displayed event probabilities are lower bounds; a rerolled-tie result instead receives a conservative conditional error bound. Displayed means and deviations use the retained distribution, and their error is not bounded by the probability-mass bound.
The fraction beside a result is a readable rational approximation with denominator up to 10,000, not a proof of exactness. Very small probabilities can be shown with a threshold label; CSV retains the calculated floating-point value.
The implementation has automated reference checks against independently generated small-case probabilities, as well as enumeration, mass, monotonicity, and error-handling checks. These checks do not establish every arbitrary game rule or every device is supported.
Mathematical reference: NIST: binomial distribution. For a more general programming environment, see AnyDice documentation or dice.run expressions. DiceClarity is independently implemented and is not affiliated with these projects.
Original calculation · Supported within the stated scope · DiceClarity 1.0 tested configurations only; not every game mechanic.
No real-user usability or preference testing. Infinite tails are truncated with an explicit probability bound; arithmetic is floating point.
| Rule | Boundary |
|---|---|
| Ordinary dice / pools | 1–100 dice, 2–100 sides; an operation budget also applies. |
| Custom faces | 1–100 equally likely entries, integers −100 to 100. |
| Keep highest / lowest | At most 20 dice and 20 distinct faces; sum/custom models. Large states may still exceed the budget. |
| Reroll the lowest | At most 1.5 million enumerated outcomes; no other per-die special rules. |
| Explosions | Ordinary sums and non-negative success pools. No custom faces, cancellation, per-die rerolls, keep/drop, or whole-roll retry. |
| Mixed dice | Up to 8 additive terms and 100 dice; whole-roll repeats and a fixed bonus are available. |
| Inverse search | 1–100 dice; success pools, at-least target; no selective or whole-roll failure retry. |
Calculations run in a separate browser worker with a work budget and an eight-second timeout. On an error, inputs remain editable. Reduce the count or simplify the rule and calculate again. Unsupported combinations are rejected rather than interpreted as a different rule.
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