Let $A_N$ be an $N \times N$ matrix whose entries are i.i.d. Bernoulli random variables with probability $1/2$, i.e.,

$$\mathbb{P}( (A_N)_{ij} =0) = \mathbb{P}( (A_N)_{ij} =1) = \frac{1}{2}.$$

Let $p_N$ be the probability that $\det A_N$ is odd. Find $\lim_{N \to \infty} p_N$.