Ways of getting a number with $n$ dice, each with $k$ sides
Assume the dice are numbered from $1$ to $k$. My hunch is that this will form a normal distribution with a median at $n\cdot\frac{k}{2}$. However, I have no idea as to
- turn this fact into an answer (I have a minimal knowledge of stats, but I know that I am missing the standard distribution)
- and this is probably the wrong approach
How can I approach and solve this problem? (Aside, this is not for a class, stats or other, so any and all approaches welcome).
*Edit: * I want to find the number of ways that the sum of the numbers that are rolled has a particular value, if $n$ dice are rolled, and each has $k$ sides, numbered $1$ to $k$.
The exact probability is a bit complicated: the chance of getting a total of $p$ when you throw $n$ $k$-sided dice is: $${1\over k^n}\sum_{j=0}^{\lfloor (p-n)/k\rfloor} (-1)^j {n\choose j}{p-kj-1\choose n-1}$$ You can see an explanation and examples here, see equation (10).
You are correct that for moderately large $n$ the distribution is well approximated by a normal curve with mean $n(k+1)/2$ and standard deviation $\sqrt{n(k^2-1)/12}$.