Buffon's Needle Probability and Regularity of Sets

Given a one-dimensional set in the plane, what is the likelihood that a random line intersects the given set? This probability is fundamentally connected to the regularity of the given set. In particular, if a set is similar to a finite length curve, then the probability is positive. Moreover, the probability is zero if and only if the set meets every finite length curve on a null set. We will discuss this classical regularity result and related questions pertaining to fractal sets.