Dice Probability Calculator
Pick your dice and a target sum to get the exact, at-least and at-most probabilities — as a fraction, decimal and percent — plus the full sum-distribution chart and step-by-step working.
Dice probability explained, step by step
Every face of a fair die is equally likely, so probability comes down to counting favorable outcomes and dividing by the total. Each card below covers one piece of the counting logic this calculator uses.
1. Sample space — every face is equally likely
With one s-sided die each face has probability 1/s. Roll n independent dice and the number of ordered outcomes multiplies to s^n. Two d6 have 6² = 36 ordered outcomes; three d6 have 6³ = 216.
2. Count the ways to make a target sum
P(sum = T) is (ways to make T) ÷ s^n. The ways come from the coefficient of x^T in (x + x² + … + x^s)^n. The calculator builds these counts with a small dynamic-programming loop — exact even for 6d20.
3. At-least and at-most probabilities
P(sum ≥ T) sums ways for every T′ ≥ T; P(sum ≤ T) sums the other tail. Because probabilities on disjoint outcomes add, you never double-count as long as ranges don't overlap.
4. Why sums look triangular then bell-shaped
One die is uniform, two dice give a triangle (many ways to make middle sums, one way for extremes), three or more dice quickly look like a bell — a direct visual of the Central Limit Theorem for independent identical variables.
Features of this calculator
- Supports 1–6 dice with any of the six standard polyhedral shapes: d4, d6, d8, d10, d12 and d20.
- Computes exact, at-least and at-most probabilities in one click — no rounding, no simulation.
- Shows every probability three ways: reduced fraction, decimal and percent.
- Draws the full probability distribution of the sum with the target range highlighted.
- Show/hide step-by-step working — including how many face-combinations make each sum.
- Copy the summary as text or download the result panel and chart as a PNG.
Frequently asked questions
+What is the most probable sum with 2d6?
7, with probability 6/36 = 1/6 ≈ 16.67%. It's the only sum every face of one die can pair with a valid face on the other die.
+What are the odds of rolling snake eyes (two 1s)?
Exactly one favorable outcome out of 36 — P = 1/36 ≈ 2.78%. Same as rolling boxcars (two 6s).
+How is this different from a dice roller?
A dice roller simulates individual rolls at random. This calculator returns the true probability, computed exactly from the underlying combinatorics — no sampling error, no simulation runs needed.
+Does this handle unusual dice like d3 or d100?
Not yet — the dropdowns cover the six standard polyhedral shapes (d4, d6, d8, d10, d12, d20) that account for almost every real-world use. The underlying algorithm supports arbitrary sides; ping us if you need d100 support and we'll add it.
+Are the probabilities exact or approximate?
Exact. The calculator uses BigInt arithmetic to count outcomes and only rounds when presenting the decimal and percent form. The fraction is always fully reduced.
+How do I read the distribution chart?
Each bar is a possible sum. Bar height is the number of ways to roll that sum (equivalently, the probability, since every ordered outcome is equally likely). Bars matching your query (exact, ≥ target, ≤ target) are highlighted in the accent color.