You might be wondering: what are the chances of rolling exactly a 4 with a pair of two regular 6-sided dice?

Well, this is clearly a probability problem that requires some maths. But before we go to complex formulae, why don’t we try a thought experiment instead:

In our problem above, we can start by thinking about what are all the possible sums we can roll. With such a pair of two regular 6-sided dice, we can only ever hope to roll a 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, or 12.

Next you have to realize that these results are not equivalent! For example, there is only one (!) way you can obtain a 2: by rolling a 1 and then another 1, but to roll a sum of 4 there are three (!) different possibilities: 1+3, 2+2, 3+1. Similarly, to roll a sum of 12 there is, once again, only one possibility: 6+6.

You can write out all the possibilities for each result, and you will find that 7 is the most likely result with six (!) possibilities: 1+6, 2+5, 3+4, 4+3, 5+2, 6+1.

You will also find that the total number of rolling the dice in different ways is 36. I.e. the number of permutations with repetitions in this set is 36. We can therefore calculate the probability of rolling a target sum as the ratio of our targeted sum to all possible outcomes:

In our earlier example, rolling exactly a 4 with a pair of two regular 6-sided dice, the probability thus is Proability(4) = 3 possibilities / 36 total possibilities = 3 / 36 = 1 / 12 = 8.33%.

If you have more dice, then this quickly becomes almost impossible to calculate by hand. This is where maths comes to the rescue. We define the general problem of calculating the probability of rolling an exact sum “r” out of a set of “n” “s”-sided dice.

The formula to calculate the probability then is:

This will calculate the probability of rolling an exact sum “r” out of any number “n” of “s”-sided dice you might have. For regular 6-sided dice, you can replace the “s” in the formula with the number 6.