Geometric probability of intersection of a square and a circle

Question

In the unitary square we choose a point $(X, Y)$ with iid coordinates $U [0,1]$ and a radius $R$, independent of $(X, Y)$ and $U [0,1]$, and we draw the circle of radius $R$ with center $(X, Y)$. Find the probability that this circle intersects the circumference of the unit square.

Answer 1

In order to not intersect for given $R$, the center must be in a smaller square area $(1-2R)^2$ (which is course only possible when $R<\frac12$). Hence we have intersection with probability $$1-\int_0^{\frac12}(1-2r)^2\,\mathrm dr. $$