Hitting probabilities for non-convex lattice

Caristi and Stoka [7] and [8] introduced in the Buffon-Laplace type problems so-called obstacles. They considered two lattices with axial symmetry and in a first moment [7] they study with eight triangular and circular sector obstacles and in the second moment [8] they analyze twelve obstacles. Several other authors considered different lattices with different types of obstacles and studied the probability that a random body test intersect the fundamental cell [2, 5], and [4]. In particular, in [1], the authors studied a Laplace type problem with obstacles for two Delone hexagonal lattices and in [6] for a regular lattice of Dirichlet-Voronoi. In this study, starting from the results obtained by Duma and Stoka [9] for Buffon type problems with a nonconvex lattice we consider a Laplace type problem for three lattices with triangular obstacles, circular sector obstacles and triangular and sectors circular together. We study the probability that a random segment of constant length intersects the fundamental cells in Figures 1, 3, and 5.