The probabilities for the number of intersections in the Buffon-Laplace needle problem in $\mathbb{R}^d$

Abstract

In 1974, Stoka solved Buffon's needle problem in $\mathbb{R}^d$, $d \ge 2$, i.e. he found a closed form solution for the probability that a line segment ("needle") with length $\ell$ intersects a grid of parallel hyperplanes with mutual distance $a\ge\ell$. For the Laplace needle problem in $\mathbb{R}^d$, where there are $d$ families of parallel hyperplanes with distances $a_1,\ldots,a_d$ fulfilling $\min(a_1,\ldots,a_d)\ge\ell$, and normal vectors in the direction of the coordinate axes $x_1,\ldots,x_d$, he was only able to give a closed solution for the case that the needle intersects hyperplanes of all families simultaneously. In the present paper, we calculate the probabilities $p_d(i)$ of exactly $i$, $0\le i\le d$, intersection points between the needle and the hyperrectangular grid formed by the $d$ families, and conclude the expected value and the variance for the number of intersection points. Furthermore, we present a simulation program and some numerical results.