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.
Last updated: September 2026
Formula below · 2 sources (mathworld.wolfram.com, Wikipedia) · Updated Sep 2026
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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 builds the exact distribution of the sum by convolving one die at a time (each face adds 1/s of the probability), then adds up the sums that meet the comparison condition (equal to, at least, or at most the target). This equals favorable outcomes / sⁿ exactly and works for up to 100 dice.
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 than or equal to' to find the chance of rolling 7 or more (21/36 ≈ 58.3%).
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 select 'Greater than or equal to' directly: for two d6 wanting at least 8, set target = 8 and the calculator counts the 15 favorable outcomes, giving 15/36 ≈ 41.7%. Equivalently, 'Less than or equal to' on 7 gives 21/36 ≈ 58.3%, and 1 − 0.583 = 0.417.