Dice Probability Calculator
Calculate the probability of rolling a target sum with a given number of fair dice.
Inputs
P(sum = target)
0.166667
Favorable Outcomes
6
Total Outcomes
36
Step by step
Values used
Number of Dice = 2; Sides per Die = 6; Target Sum = 7
Formula applied
P(sum=t) = (1/s^n) × Σ (-1)^k × C(n,k) × C(t−n−ks+n−1, n−1)
P(sum = target)
= 0.166667
Favorable Outcomes
= 6
Total Outcomes
= 36
How it works
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
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.