Given a matrix with elements from a finite field $\mathbb{F}_q$, $A\in\mathbb{F}_q^{N\times M}$, where $q$ is the size of the field, $N<M$. Suppose that $A$ in the reduced row echelon form. Obviously, the rank of such $A$ is $N$.
Suppose that now I generate a $M$-dimensional random vector $\mathbf{f}\in\mathbb{F}^{1\times M}$, with elements uniformly randomly chosen from $\mathbb{F}_q$. Is that possible to exactly compute the probability that $\mathbf{f}$ is linearly independent with those $N$ rows of $A$? I feel like it should be close to $1-\frac{q^N}{q^M}$, but not much sure. If it is not, any clue to look into the problem? Thanks.
Yes, it is possible to exactly compute, and we need only classical probability. Your question amounts to choosing an $N$-dimensional subspace of $\mathbb{F}_q^M$ (the row space of your given matrix which you said has rank $N$) and then asking the probability that a randomly chosen vector does not lie in that subspace. Well since the subspace contains $q^N$ elements and the whole space contains $q^M$ elements, and each vector is equally likely to be selected, the probability is $$ \frac{q^M -q^N}{q^M} = 1 - \frac{q^N}{q^M} $$ as you suspect.