Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
What information about the dice are you tracking? The faces? The sum? – Austin Mohr May 29 '13 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. – Anubian Noob Jun 1 '13 at 15:27
up vote 7 down vote accepted

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 $\frac{35}{12}$.

share|cite|improve this answer
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 – Anubian Noob May 31 '13 at 0:51
@user2197700: I have added a section on the mean and variance of a single die. – robjohn May 31 '13 at 11:06
Ok thanks! I can figure out the function from that! Now I just need 7 more rep to upvote XD – Anubian Noob Jun 1 '13 at 15:26

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.