Dice Roll Probability Calculator
Compute the probability of rolling any target sum with any number of multi-sided dice. Perfect for tabletop RPG designers, board game players, and statistics students.
About this calculator
When rolling n dice each with d sides, the total number of equally likely outcomes is d^n. The number of ways to reach a specific sum s is found by counting all combinations of n integers (each between 1 and d) that add up to s — a classic combinatorial problem solved by recursive enumeration. The probability is then: P(sum = s) = countWays(n, s, d) / d^n. Valid sums range from n (all ones) to n×d (all maximums). Sums near the middle are most probable because many combinations produce them, while extreme sums (minimum or maximum) have only one combination each, making them the rarest outcomes.
How to use
Roll 2 six-sided dice and find the probability of summing to 7. Enter numDice = 2, targetSum = 7, sides = 6. Total outcomes = 6^2 = 36. The ways to make 7 are: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) = 6 ways. Probability = 6/36 = 0.1667, or 16.67%. This is the most likely single sum when rolling 2d6, which is why 7 is the pivotal number in many dice games.
Frequently asked questions
What is the most probable sum when rolling two six-sided dice?
The most probable sum is 7, achievable in 6 out of 36 combinations (about 16.67%). Sums decrease symmetrically on either side: 6 and 8 each have 5 ways, 5 and 9 have 4 ways, and so on down to 2 and 12 which each have only 1 way. This bell-shaped distribution is why many board games hinge on the number 7.
How do I calculate dice roll probability for non-standard dice like d8 or d20?
Simply enter the appropriate number of sides in the Sides per Die field. The formula works for any die shape: d4, d8, d10, d12, d20, or even non-standard dice like d3. The total outcome space becomes d^n, and the recursive counting logic finds every valid combination automatically, regardless of the die type.
Why does adding more dice make the sum distribution look like a bell curve?
This is a direct consequence of the Central Limit Theorem. Each die contributes an independent, uniformly distributed value. When you sum many independent random variables, the distribution of their total converges toward a normal (bell-shaped) distribution. With just 2 dice you already see a triangular peak; by 4 or more dice the curve is noticeably smooth and symmetric around the expected mean of n × (d+1)/2.