Probability density for related variables

Homework Statement

Say I calculated a probability density of a system containing m spins up (N is the total number of particles). The probabilities of being up and down are equal so this is easy to calculate. Let's call it $\omega_m$. Then we define magnetization as $M=2m-N$ and it asks me to calculate the probability density of M.

Homework Equations

$\omega_m(m)=\frac{1}{2^N}\frac{N!}{m!(N-m)!}$

The Attempt at a Solution

I'm not sure how to interpret a probability density of M. M can have a value between -N and N, so the probability, for example, of M=-N is the same as m=0. This suggests me that I can simply replace m for (M+N)/2 in its probability density expression. I don't know if that makes sense, the mathematical properties of these probability densities are no where to be found (at least not with this particular detail).

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I think my comprehensioon of that probability density improved a bit. It's the probability of the system to be in the state defined by N and m (or M), so my solution obtained from simply rewriting $\omega$ as $\omega(N,M)$ is right. Right?

haruspex