We will generate a random line in the hyperbolic plane. Working in the Poincaré disk model, choose to random points on boundary of the disk. These will correspond to Ideal points in the hyperbolic plane. We can then draw a line between the points, forming a line in the hyperbolic plane.
(Sidenote: This probability distribution is not absolute, but rather depends on which point you choose as the center of the Poincaré disk model.)
What is the expected value of the distance between this line, and the point corresponding to the center of the disk (assuming the curvature is $-1$)?
(By distance, I mean the length (in the hyperbolic plane) of the line segment connecting the line and point, such that the line segment is perpendicular to the line.)