Probabilidade Soma de Dados

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.