Bertrand's paradox on a monitor

We investigate Bertrand’s probabilistic paradox through the lens of discrete geometry and old-fashioned but reliable discrete probability. We approximate the plane unit circle with $1\mathrm{/}n$ times $1\mathrm{/}n$ boxes and count the pairs of boxes separated by distance more than $\sqrt{3}$. For $n\to \mathrm{\infty }$ the proportion of such pairs goes to $$\frac{1+\sqrt{3}}{8}-\frac{\pi (2-\sqrt{3})}{96}=0.33273\dots \mathrm{.}$$