What is the probability that a matrix over $GF(2)$ fails to have full row-rank?

Question

This question relates to this post, which is ultimately concerned with this paper. I do my best to relay the relevant information here.

Within the linked paper, the authors seem to work under the assumption that if we select a random $288 \times 216$ matrix $M$ with entries in $GF(2)$ (the field with two elements), then there is a very high probability that $M$ has full column rank, and that a randomly selected $84 \times 216$ submatrix will have full column rank. With that in mind, my question is the following:

If a random $m \times n$ matrix $M$ over $GF(2)$ is selected (i.i.d. uniformly random $\{0,1\}$ entries), then what is the probability that $M$ will have maximal rank in the case where

1. $m = 288, n =216$
2. $m = 84, n = 216$

I am aware that there is a nice formula for this probability in the case where $M$ is square, but I'm not sure where to find (or how to derive off the top of my head) the corresponding probability for rectangular matrices.

Any help is appreciated.

Answer 1

Following the formula given here, we find that the answer to 1 is $$ \prod_{i=1}^{216}(1 - 2^{i-1-288}) \approx 1 - 2\times 10^{-22}, $$ and the answer to 2 is $$ \prod_{i=1}^{84}(1 - 2^{i-1-216}) \approx 1 - 2\times 10^{-40}. $$