A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0), (2020, 0), (2020, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{2}$. (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integers.) What is $d$ to the nearest tenth?
This is a typical geometric probability problem. A point within $d$ units of a lattice point must lay within a circle centered at that lattice whose radius equals $d$. If we draw all such circles on the squares and then the probability will be the sum of these circles' areas divides the area of the square.
If we consider each unit square, it is clear that it will have a $4$ quarter-circles within that square, therefore the ratio of the ares must be $\pi d^2 : 1$. This means $$\pi d^2 =\frac{1}{2} \implies d=\frac{1}{\sqrt{2\pi}}\approx \boxed{0.4}$$
Note: geometric probability is discussed in the book Counting.