Dice Probability Calculator

Dice Probability Fundamentals

When you roll a single fair die with s sides, each face has an equal probability of 1/s. With multiple dice, outcomes combine according to probability theory.

Exact Probability Distribution For n dice each with s sides, the probability of a specific sum k is computed using convolution: repeatedly combine the probability distribution of one die with the running total. The minimum sum is n (all ones), the maximum is n × s (all maximum faces).

Mean and Standard Deviation The expected (average) sum for n dice with s sides is: Mean = n × (s + 1) / 2. For example, 3d6 has mean 3 × 3.5 = 10.5. The standard deviation is: σ = √(n × (s² − 1) / 12).

Advantage and Disadvantage (D&D) Advantage: roll 2d20 and take the higher result. P(result ≤ k) = (k/20)². Disadvantage: take the lower result. P(result ≤ k) = 1 − ((20−k)/20)². These significantly shift the probability distribution.

At Least / At Most Calculations P(sum ≥ target) = sum of P(sum = k) for all k ≥ target. P(sum ≤ target) = sum of P(sum = k) for all k ≤ target.

At Least One Specific Face The probability of rolling at least one 6 on n dice: P = 1 − (5/6)ⁿ. More generally: P = 1 − ((s−1)/s)ⁿ for the highest face.

Large Dice Pools For large numbers of dice, the central limit theorem applies — the distribution approaches a normal (bell curve) distribution with the mean and standard deviation above.