Probability that a random binary matrix will have full column rank?
Let $m>n$, and $\{v_1,\ldots,v_{n+1}\}$ be random vectors in $\mathbb{F}^m_2$. Call $p_n$ the probability that $\{v_1,\ldots,v_n\}$ are linearly independent.
In that case, $\{v_1,\ldots,v_{n+1}\}$ are independent (event $A_n$) if and only if $\{v_1,\ldots, v_n\}$ are independent and $v_{n+1} \notin \left<v_1,\ldots,v_n\right>$ (event $B_n$).
$$p_{n+1} = P(A_n \cap B_n) = P(A_n)P(B_n|A_n) = (1-2^{n-m})p_n$$
$p_1 = (1-2^{-m})$, since a single vector is independent if and only if it's not $0$, so: $$p_n = \prod_{i=1}^n (1-2^{i-1-m})$$