Problems of Buffon type for large test bodies (Q2772811)

In the classical formulation of the problem of Buffon-Laplace type, a ``small'' convex test body \(T\) is thrown at random onto a lattice of parallel and equidistant lines, or of rectangles, and one asks for the probability of an intersection. The request that \(T\) be ``small'' assures that \(T\) has at most one intersection with the lattice. When this condition is not satisfied, i.e. for ``large'' test bodies, there can be multiple intersections and problems connected with the probability of a specific number of crossings arise. These problems have been studied by various authors (we mention, among others, Diaconis, Duma and Stoka), but usually working with planar lattices.NEWLINENEWLINENEWLINEIn this paper the author investigates the three-dimensional case, using a lattice of parallel planes and a needle and a circle as test bodies. The work is well written and contains original and very interesting results about the distribution of the number of crossings and its approximate moments. In particular the following theorem is noteworthy: Let us consider a lattice of parallel planes, at a distance \(d\) apart, and a disk \(D\), of radius \(r\), \(nd\leq 2r\leq (n+ 1)d\), with random position in \(E_3\). We denote by \(L\) the random variable ``sum of the lengths of the chords of \(D\) resulting as intersection between \(D\) and the planes of the lattice''. The mean value of \(L\) is \(E[L]= \pi r^2/(4d)\).