Let $n\leq m$ and $0\leq k\leq (n\times m - \min\{n,m\})$ be in $\mathbb{N}$. Let $\mu$ be a probability measure dominated by the Lebesgue measure on $\mathbb{R}$ and generate a random $n\times m$ matrix $A$ as follows: $$ A_{i,j} \sim \mu $$ and set exactly $k$ elements of $A$ equal to $0$ where the indices of the zero entries are chosen uniformly at random.

What is the probability that $A$ has full rank?

For example, if $n=m$, $k=0$ and $\mu$ is Gaussian then $det(A)\neq 0$ with probability $1$. So $A$ has full rank... but what about in general?