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 is P = 1 − (5/6)ⁿ. The same formula works for any single face you name, not only the highest: P = 1 − ((s−1)/s)ⁿ. Rolling six d6 gives you a 66.5% chance of at least one 6, which is well short of the certainty most people expect from six chances at a one-in-six shot.

Advantage is worth the most in the middle The gain from advantage is not constant. Against a target of 2 it adds almost nothing, because you were going to make it anyway. Against a target of 20 it takes you from 5% to 9.75%, which doubles your chance but is still a long shot. The peak is at 11, where a flat d20 gives 50% and advantage gives 75%, a full 25 percentage points. That is the number to remember: advantage is worth about +5 on the die at the extremes and closer to +5 on the roll in the middle.

Disadvantage mirrors it exactly. The average of the higher of 2d20 is 13.825, the average of the lower is 7.175, and the two sum to 21 for the same reason a single d20 averages 10.5.

Large Dice Pools For large numbers of dice the central limit theorem takes over and the distribution converges on a normal curve with the mean and standard deviation above. By 5d6 the bell shape is already close enough that a normal approximation is within a percentage point almost everywhere, which is why table-top designers reach for 3d6 and 5d6 when they want outcomes to cluster instead of spread flat.