On the singularity of random Bernoulli matrices - novel integer partitions and lower bound expansions derives the lower bound (theorem 1) $$\mathbb{P}\{ \text{det}(A_n) = 0\} \geq 2^{2-n} \binom{n}{2}-2^{-2n} \left(12 \binom{n}{2}^2-4 \binom{n}{2}\right),$$ which is of order $n^2 2^{1-n}$ for $n\rightarrow\infty$.

On the singularity of random Bernoulli matrices - novel integer partitions and lower bound expansions derives the lower bound (theorem 1) $$\mathbb{P}\{ \text{det}(A_n) = 0\} \geq 2^{2-n} \binom{n}{2}-2^{-2n} \left(12 \binom{n}{2}^2-4 \binom{n}{2}\right),$$ which is of order $n^2 2^{1-n}$ for $n\rightarrow\infty$. The "independent selection" contribution is the first term $2^{2-n} \binom{n}{2}$. The second term is smaller by a factor $n^2/2^n$ for large $n$.