Consider two independent and identically distributed random vectors of dimensionality $N$, $\mathbf{x}$ and $\mathbf{y}$ where their elements are iid generated from Gaussian with zero-mean and variance equal to $\sigma^2$, i.e., $\mathbf{x} \& \mathbf{y} \sim \mathcal{N}(0, \sigma^2)$.

What is the probability that $\mathrm{sign}(\mathbf{x})=(\mathrm{sign(x_1)},\mathrm{sign(x_2)}, \ldots, \mathrm{sign(x_N)})$ be equal to $\mathrm{sign}(\mathbf{y})=(\mathrm{sign(y_1)},\mathrm{sign(y_2)}, \ldots, \mathrm{sign(y_N)})$ if their Euclidean distance is smaller than $r$, $r \geq 0$: $\Pr\left[ \mathrm{sign}(\mathbf{x})=\mathrm{sign}(\mathbf{y})\ \vert\ \Vert \mathbf{x} - \mathbf{y} \Vert \leq r\right]$.

Note: $\mathbf{x}$ and $\mathbf{y}$ are independent.