Banach's matchbox problem

Question

A mathematician keeps a matchbox in each of two pockets, each box starting with $n$ matches. Whenever he wants a match he reaches into a pocket chosen uniformly at random and takes one. Eventually he reaches into a pocket, pulls out the box, and finds it empty.

At that moment, what is the probability that the other box contains exactly $k$ matches? As a follow-up, what is the expected number of matches remaining in the other box?

Accepted Answer

Fix which box is discovered empty, say the left one, and multiply by 2 at the end by symmetry. For the left box to be found empty while the right box holds exactly k matches, the sequence of pocket selections up to and including the discovery must consist of: - all n successful draws from the left box, - exactly n-k draws from the right box (leaving k there), - and finally one more selection of the left pocket, the reach that finds it empty. So there are n + (n-k) = 2n-k selections before the final one, and the final selection is left. Among those first 2n-k selections, exactly n were left (i…

Banach's matchbox problem

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