Expected number of intersections of a line piece dropped on a set of random line pieces of the same length Consider line pieces of length $L$ distributed on the plane with random orientations and their centers of mass randomly distributed at a spatial density $n$. Drop another line piece with the same length on this ensemble. What is the expected number of intersection our probe line piece has with the other line pieces? Intuition, and dimensional reasoning, suggests this number will be proportional to $nL^2$. How could I prove this? Is there a prefactor?
 A: This reduces to a simple case of Buffon's Needle problem.  Let $\mathcal L$ be the last segment dropped onto the plane.  It determines two translations:

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*$\sigma_{\mathcal L}$ moves the plane a distance $L$ orthogonal to $\mathcal L;$


*$\tau_{\mathcal L}$ moves the plane a distance $L$ parallel to $\mathcal L.$
The union of all the translates $\sigma_{\mathcal L}^m\tau_{\mathcal L}^n\, \mathcal L$ for integers $m,n$ is a set of parallel lines $\mathbb L$ spaced by $L:$ the setting of Buffon's Needle problem.  Thus, when there are an average of $\rho$ centers of mass per $L^2$ area, the rate at which the initial random segments intersect one of these parallel lines is $2\rho/\pi.$
The translates $$\mathbb{L_2} = \{\sigma_{\mathcal L}^m\tau_{\mathcal L}^{2n}\,\mathcal L \mid m, n\in \mathbb Z\}$$ (with even powers of $\tau$) occupy half of these lines and the omitted half are just the $\tau$-translates of $\mathbb{L_2}.$  In this figure, $\mathbb{L_2}$ is drawn in solid gray while $\tau\,\mathbb{L_2}$ is drawn with dashes.  $\mathcal L$ itself is shown in red, with the translations $\sigma$ and $\tau$ shown as vectors.

Since the random line segment process is homogeneous (in location and orientation), we may view it as a process on the torus $\mathbb{Z}^2 / \sigma\tau^2.$  The problem concerns the rate at which the random line segments intersect (the image of) $\mathcal L$ in this torus.
But because the chance that any line segment intersects both $\mathcal L$ and $\tau\,\mathcal L$ is zero, the rate of intersection with $\mathbb{L_2}$ is half the rate of intersection with $\mathbb{L};$ that is, it is $\rho/\pi.$
