Dice Probability Calculator
Calculate the probability of rolling at least, exactly, or at most a target sum with N dice of S sides.
What this calculates
Rolling N dice produces a triangle/bell-shaped distribution, not the flat curve of one die. This calculator enumerates all possible outcomes and returns the probability of hitting any sum threshold — handy for tabletop RPGs, board game strategy, and probability homework.
Formula & how it works
Total outcomes = S^N. Count outcomes matching condition via dynamic programming over the sum distribution. P = count ÷ total. Mean = N × (S+1)/2; variance = N × (S²−1)/12.
Worked example
Roll 4d6 (drop none). P(sum ≥ 18) ≈ 13.27%. Mean sum = 14. Min 4, max 24.
Frequently asked questions
Does this handle d20 or d100?
Yes — set sides to 20 or 100. Calculation slows with very large N×S but stays accurate.
Advantage/disadvantage?
Not in this version. For 2d20 keep highest (advantage), expected value rises by ~3.3.
Why bell-shaped?
Central limit theorem. As N grows, the distribution of sums approaches a normal curve regardless of die shape.