  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 $1/n$ times $1/n$ boxes and count the pairs of boxes
separated by distance more than $\sqrt{3}$. For $n\to\infty$ the proportion of
such pairs goes to $$
\frac{1+\sqrt{3}}{8}-\frac{\pi(2-\sqrt{3})}{96}=0.33273\dots\;. $$
