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Dice probability calculator

Find the probability that identical fair dice total exactly, at most, or at least your target sum.

Inputs

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Whole number from 1 to 20.

Each fair die is numbered 1 through this value.

The target must be between the minimum and maximum possible sums.

Result

Enter values and calculate to see the result.

How to use this dice probability calculator

Choose how many identical fair dice you will roll and how many numbered sides each die has. Then select whether the total must equal, stay below, or reach your target.

  1. Enter 1–20 dice.
  2. Enter 2–20 sides per die.
  3. Choose exactly, at most, or at least.
  4. Enter a target within the displayed possible sum range, then select Calculate.
  5. Edit any input to clear the confirmed result before a new calculation.

How the calculation works

One fair s-sided die gives each face probability 1/s. The calculator starts with total 0 before any roll and adds one die at a time. This finite dynamic-programming table is equivalent to reading coefficients of (x + x² + … + xˢ)ⁿ, divided by sⁿ.

Exactly uses one table entry. At most adds every probability from the minimum through the target. At least adds from the target through the maximum. The main probability is exact; the small random trials above are illustrative only.

Worked examples

Two six-sided dice total 7

Six of the 36 equally likely ordered outcomes total 7, so the exact probability is 1/6, or 16.67%.

Two six-sided dice total at least 10

Totals 10, 11, and 12 have 3 + 2 + 1 favorable outcomes. The probability is also 6/36 = 16.67%.

Input limits and interpretation

  • The dice are independent, fair, and have the same number of sides.
  • Faces are consecutive integers from 1 through the selected side count.
  • The smallest sum is the number of dice; the largest is dice × sides.
  • Displayed percentages use at most two decimals; very small nonzero values use scientific notation.
  • This is a theoretical probability, not a promise about a short series of real rolls.

Frequently asked questions

Why is 7 more likely than 2 with two d6?

Six ordered pairs total 7, but only (1,1) totals 2.

Are “at least” and “more than” the same?

No. At least includes the target; more than would begin at the next integer.

Can I mix different dice?

No. This page covers identical dice only.

Are the sample rolls used for the probability?

No. They are a small random illustration; the displayed probability comes from the exact finite distribution.