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

TL;DR

This paper extends Buffon's needle problem in higher dimensions by calculating the probabilities of exactly i intersections with a hyperrectangular grid, providing explicit formulas, expected values, variances, and supporting simulations.

Contribution

It derives explicit probabilities for the number of intersections in the Buffon-Laplace needle problem in $\

Findings

Calculated probabilities for each number of intersections

Derived the expected value and variance of intersection counts

Provided simulation results supporting theoretical findings

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.