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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.

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    $\begingroup$ What information about the dice are you tracking? The faces? The sum? $\endgroup$ May 29, 2013 at 23:34
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    $\begingroup$ @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. $\endgroup$ Jun 1, 2013 at 15:27

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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 $\sigma=\sqrt{50\times\frac{35}{12}}$.

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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 $\frac{35}{12}$.

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  • $\begingroup$ 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 $\endgroup$ May 31, 2013 at 0:51
  • $\begingroup$ @user2197700: I have added a section on the mean and variance of a single die. $\endgroup$
    – robjohn
    May 31, 2013 at 11:06

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