Dice Probability Calculator
Exact probability that a roll of up to 20 dice with 2 to 100 sides plus a modifier is at least, exactly or at most a target, with mean, spread and percentiles.
Interactive calculator · Method DICE-PROBABILITY-1.0 · Last reviewed: 2026-11-04
Calculate your scenario
Each of the sidesⁿ outcomes is equally likely for fair dice.
Your inputs
Default worked example. Edit the inputs to calculate your scenario.
Default worked example
3d6: 135 of 216 equally likely outcomes give a total at least 10, so P = 62.5%; mean = 3 × (6 + 1) ÷ 2 = 10.5.
How to use the calculator
Enter how many dice and how many sides each has, any flat modifier, the target total and whether you need at least, exactly or at most that total.
What the result means
With 3d6, 135 of the 216 outcomes total 10 or more, so the chance is 62.5% (1 in 1.6); 10 and 11 are the most likely totals and the average is 10.5.
Formula and units
Each of the sidesⁿ outcomes is equally likely for fair dice. The number of ways to reach each sum is built exactly by convolving one die at a time (whole-number arithmetic, no simulation); the modifier shifts every total. Probability = ways meeting the condition ÷ sidesⁿ; odds 1 in N with N = sidesⁿ ÷ ways. Mean = n × (sides + 1) ÷ 2 + modifier; standard deviation = √(n × (sides² − 1) ÷ 12).
Worked examples
3d6, total at least 10: Number of dice: 3 dice; Sides per die: 6 sides; Modifier added to the total: 0 dimensionless; Target total: 10 dimensionless; Condition: atleast.
Example result: Probability: 62.5%; Odds (1 in N): 1 in 1.6; Mean total: 10.5; Standard deviation: 2.958; Range (min – max): 3 – 18; Most likely total(s): 10 and 11.
1d20 + 5, total at least 15: Number of dice: 1 dice; Sides per die: 20 sides; Modifier added to the total: 5 dimensionless; Target total: 15 dimensionless; Condition: atleast.
Example result: Probability: 55%; Odds (1 in N): 1 in 1.82; Mean total: 15.5; Standard deviation: 5.7663; Range (min – max): 6 – 25; Most likely total(s): every total from 6 to 25 (equally likely).
Limitations and common mistakes
Assumes fair, independent dice numbered 1 to the number of sides, summed. It does not model rerolls, dropping dice, exploding dice or loaded dice. A probability describes long-run frequency; it does not predict any particular roll, and no betting system changes the odds.
Adding single-die probabilities (or assuming every total is equally likely) for several dice; middle totals are far more likely than the extremes.
Sources and assumptions
Method DICE-PROBABILITY-1.0; last formula review 2026-11-04. Sources and their supported claims. Calculation methodology.