Suppose we have a matrix $A \in \{0,1\}^{n \times n}$ where

$$A_{ij} = \begin{cases} 1 & \text{with probability} \quad p\\ 0 &\text{with probability} \quad 1-p\end{cases}$$

I would like to know the expected value

$$\mathbb E( \det (A) =0) \ \text{ for large }n,$$

i.e., the asymptotic probability of the determinant being zero as $n$ becomes large.

I know that the expected value

$$\lim_{n \rightarrow \infty} \mathbb E \left( \det (A) =0 \right) = 0$$ if $p \neq 0,1$ but could not find results for finite $n$/the asymptotic behavior.

Suppose we have a matrix $A \in \{0,1\}^{n \times n}$ where

$$A_{ij} = \begin{cases} 1 & \text{with probability} \quad p\\ 0 &\text{with probability} \quad 1-p\end{cases}$$

I would like to know the probability

$$\mathbb P( \det (A) =0) \ \text{ for large }n,$$

i.e., the asymptotic probability of the determinant being zero as $n$ becomes large.

I know that the probability

$$\lim_{n \rightarrow \infty} \mathbb P \left( \det (A) =0 \right) = 0$$ if $p \neq 0,1$ but could not find results for finite $n$/the asymptotic behavior.

Edit: I changed the symbol $\mathbb{E}$ to $\mathbb{P}$ since I meant the probability and not the expectation value as some comments assumed correctly.

Suppose we have a matrix $A \in \mathbb{N}^{n\times n}$ where

$$A_{ij} = \begin{cases} 1 & \text{with probability} \quad p\\ 0 &\text{with probability} \quad1-p\end{cases}$$

I would like to know the probability

$$\mathbb E( \det (A) =0) \ \text{ for large }n,$$

i.e., the asymptotic probability of the determinant being zero as $n$ becomes large.

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I know that the probability $$\lim_{n \rightarrow \infty} \mathbb E( \det (A) =0)=0$$ if $p\neq 0,1$ but could not find results for finite $n$/the asymptotic behavior.

Suppose we have a matrix $A \in \{0,1\}^{n \times n}$ where

$$A_{ij} = \begin{cases} 1 & \text{with probability} \quad p\\ 0 &\text{with probability} \quad 1-p\end{cases}$$

I would like to know the expected value

$$\mathbb E( \det (A) =0) \ \text{ for large }n,$$

i.e., the asymptotic probability of the determinant being zero as $n$ becomes large.

I know that the expected value

$$\lim_{n \rightarrow \infty} \mathbb E \left( \det (A) =0 \right) = 0$$ if $p \neq 0,1$ but could not find results for finite $n$/the asymptotic behavior.