I was reading a textbook and I couldn't figure out something that seemed really obvious: Assume that the space is uniformly and randomly filled with stars, and the mean radial distance between the stars is l. Photons are emitted from a stars's surface in all the directions and can collide with other stars. Now it says that the probability that a photon collides (for the first time) with a star with distance r from the star that it came from, is $$P(r) = l^{-1}e^{-r/l}$$ and did not explain why at all.
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This isn't intended to be an exact representation of the probability: it's an approximation that works reasonably well in a space with a very large number of stars that are very small compared to their distances. From the point of view of a photon (which thinks very slowly, given that it experiences no real time!) it is passing through a homogeneous "soup" of material, so you don't even need to worry about the structure of space: this is purely a one-dimensional problem (the dimension is along the direction of the photon's travel). – whuber Jul 31 '18 at 11:50