Dice Probability Calculator
Calculate the probability of rolling a target sum with a given number of fair dice.
About Dice Probability Calculator
The probability of rolling a specific sum with multiple dice is calculated using the inclusion-exclusion principle on the stars-and-bars combinatorial identity. Each die face is equally likely, so P = favorable outcomes / total outcomes.
Formula
P(sum=t) = (1/s^n) × Σ (-1)^k × C(n,k) × C(t−n−ks+n−1, n−1)
- n
- Number of dice
- s
- Sides per die
- t
- Target sum
- k
- Inclusion-exclusion index
Worked example
With the default values below, Dice Probability Calculator returns P(sum = target) of 0.166667. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Number of Diceaccepts 1–10
- 2
- Sides per Dieaccepts 2–20
- 6
- Target Summinimum 1
- 7
Results
- P(sum = target)
- 0.166667
- Favorable Outcomes
- 6
- Total Outcomes
- 36
Step by step
- 1Values used
- 2Formula applied
- 3P(sum = target)= 0.166667
- 4Favorable Outcomes= 6
- 5Total Outcomes= 36
Assumptions and limits
- The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
What is the most likely sum with 2d6?
7, with probability 6/36 ≈ 16.67%. There are 6 ways to make 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1).
Why does the number of outcomes grow so fast?
Each die multiplies the possibilities by its number of faces. Two 6-sided dice give 36 outcomes; three give 216.