Given $f_1,\ldots,f_n \in \mathbb{F}_q[x_1,\ldots,x_m]$ such that $\deg(f_i) \leq d < q$. Suppose we have for some $1 \leq j \leq m$

$$ \operatorname{trdeg}_{\mathbb{F}(x_1,\ldots,x_j)}\{f_1,\ldots,f_n\} = r$$

Let $z_1,\ldots,z_j$ be chosen uniformly and independently at random from $\mathbb{F}_q$ then what is

$$ {\Pr}_{z_1,\ldots,z_j}[\operatorname{trdeg}_{\mathbb{F}}\{ f_1(z_1,\ldots,z_j,x_{j+1},\ldots,x_m),\ldots,f_n(z_1,\ldots,z_j,x_{j+1},\ldots,x_m) \} = r ] \text{?} $$

Ideally is the probability large, i.e. not inverse-polynomial in n,m or worse? This probability would be very helpful in my research and any help is appreciated.