Three mirrors are arranged so as to resemble the three walls of a room. From the open end of this room, a ray of light enters. What is the expected length of the path taken by the ray of light before it comes out?
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What is the probability distribution of the direction of the light? For the most natural interpretation of the question, the light is shining horizontally making a random angle $\theta$ ( uniformly sampled from $(-\frac{\pi}{2},\frac{\pi}{2})$) with the normal direction of the back wall, the expected length of path diverges. – achille hui Nov 26 '19 at 17:38
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Method of images suggests that the question is equivalent to “What is the average path length of crossing the stripe of width $2d$”, where $d$ is the depth of the room.
Which is eqivalent to: $$ L = \int_0^\pi \frac{2d}{\sin\varphi}\omega(\varphi)d\varphi. $$
If the assumed distribution is uniform $\omega(\varphi)=1/\pi$, then the integral diverges.
Vasily Mitch
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