Given $f_1,...,f_n \in \mathbb{F}_q[x_1,...,x_m]$ suppose we have for some $1 \leq j \leq m$

$$ trdeg_{\mathbb{F}(x_1,...,x_j)}\{f_1,...,f_m \} = r$$

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

$$ Pr_{z_1,...,z_j}[trdeg_{\mathbb{F}}\{ f_1,...,f_n \} = r ]?$$

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.