Dice Sum Probability Calculator
Enter how many six-sided dice and a target total to get P(sum = s) by convolution, plus the raw count over 6^n. Two dice landing on 7 give 6/36.
P(soma=s)
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Probability of the sum of n dice: a classic of combinatorics
Roll 2d6 and you get 6×6 = 36 ordered outcomes, all equally likely. The sum lands somewhere between 2 and 12, but the sums don't all show up the same number of times. P(2) = 1/36 needs exactly 1+1, while P(7) = 6/36 = 1/6 can come from 1+6, 2+5, 3+4, 4+3, 5+2 or 6+1. That makes 7 the most likely sum and the peak of a triangular distribution. With nd6 you count the compositions of the target sum using parts in [1,6], which you can work out with generating functions or recursion. Push n higher and the shape drifts toward a normal distribution, courtesy of the Central Limit Theorem, with mean 3.5n and variance ≈ 2.92n.
Applications
Board games lean on this constantly. In Catan, settling near 6 and 8 pays off because those numbers come up most. Monopoly players know 7 is the common roll, which is why Jail and the third side of the board see the most traffic. D&D and other RPGs use 2d6, 3d6, or 4d6 drop-lowest to set ability scores, and Yahtzee runs on the same math. It also makes a tidy classroom example for teaching probability.
FAQ
Why is 7 the most likely sum on 2d6? It sits dead center in the [2,12] range and has the most combinations behind it, six of them. The extremes 2 and 12 each have only one way to come up.
Does the formula work for non-standard dice (d4, d8, d20)? It does. Just swap [1,6] for [1,k] when you count. The most likely sum is always whichever integer sits closest to n·(k+1)/2.
Why does 4d6 drop-lowest produce stronger D&D characters? Throwing out the lowest die pushes the whole distribution upward. The expected value climbs from 10.5 on 3d6 to roughly 12.2, and your odds of rolling 16+ on any given ability go up too.
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The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.