Dice Probability Calculator

Formula Used

For ordinary fair dice, every elementary outcome has equal probability. The calculator creates each die’s probability distribution and repeatedly combines distributions. For two independent totals, convolution adds the probability of every pair producing the same sum.

Basic probability: P(event) = favorable outcomes ÷ total possible outcomes.

Expected value: E(X) = Σ[x × P(X = x)].

Variance: Var(X) = E(X²) − [E(X)]².

At least one selected face: 1 − (1 − p)ⁿ.

Exactly k successes: C(n,k)p^k(1 − p)^n−k.

When rerolls, compound explosions, weighted keep/drop combinations, or very large state spaces make direct enumeration impractical, the calculator estimates probabilities with repeated random trials.

How to Use This Calculator

1. Enter expression A using standard notation, such as 3d6+2.
2. Choose the calculation type matching your question.
3. Set the target, pattern, success rule, or comparison expression.
4. Add optional weights, rerolls, explosions, keep rules, or drop rules.
5. Select automatic, exact, or simulation evaluation.
6. Choose simulation trials and displayed decimal precision.
7. Press Calculate probability.
8. Review the percentage, fraction, odds, statistics, chart, and distribution table.
9. Copy, print, save, or export the result as needed.

Understanding Dice Probability in Depth

Start With the Sample Space

A dice probability problem begins with its sample space: every outcome that the configured dice can produce. One fair six-sided die has six equally likely outcomes. Two fair six-sided dice have thirty-six ordered outcomes because the first die and second die can each show six values. Although totals from two through twelve are possible, those totals are not equally likely. Seven has six producing combinations, while two has only one. This calculator preserves that difference instead of treating every total as equally probable.

Exact Distributions and Convolution

For ordinary dice pools, the calculator constructs an exact distribution. It begins with a total of zero having probability one. Each new die is combined with the existing distribution. Every old result is added to every face of the new die, and their probabilities are multiplied because independent rolls occur together. Probabilities producing the same total are then added. This repeated process is called convolution and works efficiently for many standard expressions, including mixed pools such as two d6 plus one d8.

Keep and Drop Mechanics

Keep-highest, keep-lowest, drop-highest, and drop-lowest rules change which rolled dice contribute to a total. An advantage roll written as 2d20kh1 rolls two d20s and keeps the larger value. These mechanics depend on the order of individual results, so exact handling may require enumeration. The calculator uses exact enumeration when the state space is manageable and changes to simulation automatically when a requested pool would require excessive work.

Weighted Dice and Advanced Rules

A weighted die assigns different probabilities to its faces. The weights do not need to total one because the calculator normalizes them. Rerolls alter the effective distribution by replacing selected faces. Exploding dice add another roll when a threshold is reached. Compound explosions can continue repeatedly. These rules may create a theoretically unbounded total, so simulation is often the most practical approach. A safety cap prevents accidental endless chains while retaining realistic estimates.

Reading the Results

The headline percentage answers the selected question. The fraction and decimal show the same probability in alternate formats. Odds in favor compare success with failure, while odds against reverse that comparison. Expected value describes the long-run average total. Median identifies a central result, mode shows the most likely total, variance measures spread, and standard deviation expresses spread in the same units as the roll. Cumulative columns answer “at most” and “at least” questions for every possible result.

Exact Values Versus Simulation

Exact calculations are deterministic and do not vary between requests. Simulated estimates can change slightly because they use random trials. Increasing the trial count usually reduces sampling error but requires more processing. Simulation is valuable for complicated tabletop systems, unusual reroll chains, custom probabilities, and opposed mechanics. For important decisions, use a high trial count and compare the estimate with a second run. For straightforward fair dice, automatic mode normally chooses exact mathematics.

Notation Reference

Frequently Asked Questions