Hitting probabilities for random convex bodies and lattices of triangles

Abstract

In the first part of this paper, we obtain symmetric formulae for the probabilities that a plane convex body hits exactly 1, 2, 3, 4, 5 or 6 triangles of a lattice of congruent triangles in the plane. Furthermore, a very simple formula for the expectation of the number of hit triangles is derived. In the second part, we calculate the hitting probabilities in the cases where the convex body is a rectangle, an ellipse and a half disc. Already known results for a line segment (needle) follow as special cases of the rectangle and the ellipse.