Multiple Dice Probability Calculator
Calculate the probability that multiple dice produce a specific sum, or a sum greater or less than a target. Perfect for tabletop RPG players and probability students.
About this calculator
When rolling multiple dice, each combination of faces is equally likely, so the probability of an outcome is the number of favorable combinations divided by the total number of possible combinations. For n dice each with s sides, the total outcomes equal sⁿ. The calculator enumerates every possible roll by treating each trial number i (from 0 to sⁿ − 1) as a mixed-radix number in base s, extracts each die's face value, sums them, and checks whether the sum meets the comparison condition (equal to, greater than, or at most the target). The final probability is favorable / sⁿ. This brute-force enumeration is exact for up to 5 dice, covering virtually all tabletop gaming scenarios.
How to use
You want the probability of rolling exactly 7 with 2 standard six-sided dice. Set num_dice = 2, dice_sides = 6, target_sum = 7, comparison = 'equal'. Total outcomes = 6² = 36. Favorable combinations that sum to 7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) = 6 combinations. Probability = 6 / 36 = 0.1667, or about 16.67%. The calculator confirms this instantly, and you can switch comparison to 'greater' to find the chance of rolling above 7.
Frequently asked questions
Why is rolling a sum of 7 the most likely outcome with two six-sided dice?
With two six-sided dice there are 6² = 36 equally likely outcomes. The sum 7 can be formed in 6 different ways — more than any other total — because it sits in the middle of the possible range (2–12) where the most face-value combinations converge. Sums near the extremes (2 or 12) have only one combination each, making them far less probable. This is why 7 is the pivotal number in games like craps.
How does increasing the number of dice change the probability distribution of the sum?
With more dice, the distribution of sums becomes more bell-shaped due to the Central Limit Theorem: extreme totals become increasingly unlikely while values near the expected mean (n × (s+1)/2) concentrate most of the probability. For example, a single d6 has a uniform distribution, but three d6 produce a near-normal curve peaking around 10–11. This is why high-dice-count rolls in tabletop games tend to produce more 'average' results than single-die rolls.
What is the probability of rolling at least a certain sum with multiple dice?
'At least' a target sum equals 1 minus the probability of rolling strictly less than that target. In this calculator you can approximate it by selecting the 'greater' comparison on (target − 1). For two d6 wanting at least 8, set target = 7 and comparison = 'greater': favorable outcomes where the sum exceeds 7 total 15, giving 15/36 ≈ 41.7%. Alternatively, use 'less_equal' on 7 (≈ 58.3%) and subtract from 1.