# Probability Distribution of Rolling Multiple Dice

What is the function for the probability distrabution of rolling multiple (3+) dice. The function is a bell curve but I can't find the actual function for the situation. Example, what is the function for rolling 50 6 sided dice?

EDIT: A six sided die returns an integer value from 1 to 6 inclusive. I am trying to find a function to add multiple dice together.

• What information about the dice are you tracking? The faces? The sum? May 29, 2013 at 23:34
• @Austin I'm tracking the number rolled, sorry if that wasn't clear. A six sided dice gives an integer between one and six inclusive. I am trying to find a function for the sum of multiple dice. Jun 1, 2013 at 15:27

In this answer, there is a section titled "Summing Dice". It describes how convolution of the discrete function that is $$1$$ for each integer from $$1$$ through $$6$$, and $$0$$ otherwise, yields the distribution for the sum of $$n$$ six-sided dice.

Rolling $$50$$ six-sided dice will yield an approximately Normal Distribution whose mean is $$\mu=50\times\frac72$$ and whose variance is $$50\times\frac{35}{12}$$; thus, a standard deviation of $$\sigma=\sqrt{50\times\frac{35}{12}}$$. Mean and Variance of a Single Die

The mean of a single die whose faces vary from $$1$$ to $$n$$ is $$\frac{n+1}{2}$$. For $$n=6$$, this gives $$\frac72$$.

The variance of a single die whose faces vary from $$1$$ to $$n$$ is "the mean of the squares minus the square of the mean." The sum of the squares from $$1$$ to $$n$$ is $$\frac{2n^3+3n^2+n}{6}$$, so the mean is $$\frac{2n^2+3n+1}{6}$$. Subtracting $$\frac{n^2+2n+1}{4}$$ yields $$\frac{n^2-1}{12}$$. For $$n=6$$, this gives $$\frac{35}{12}$$.

• Thanks! By the way how did you get the numbers 7/2 and 35/12? I want to generalize this for any amount of any sided dice May 31, 2013 at 0:51
• @user2197700: I have added a section on the mean and variance of a single die.
– robjohn
May 31, 2013 at 11:06