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I think this problem fits better here rather than the physics stackexchange.

This is a problem that has bugged me for a long while, and might be an interesting problem for the math stackexchange.

A beam splitter is a semi-reflective pane, such as a tinted window. When you shine a light, some of it will go through and the rest reflects back.

Consider n panes arranged in series, with all planes parallel, where an incident light will pass through each pane in series. Each pane reflects a portion 1-p of incident light and lets through a portion p. The reflectivity of the panes is symmetric across both sides, and a reflection will begin going through the panes the opposite direction until reflected again. You shine a beam through the panes. Beginning at the 1st pane, how much light gets through the nth pane?

Diagram here for n=3:

I_0 --> |1| --> |2| --> |3| --> I_3

I think the best bet for a solution to this problem is recognizing that a photon bouncing through the panes is taking a random walk.

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