I had posted an [urn probability problem](https://mathoverflow.net/questions/29606/what-is-being-counted-closed) that didn't have good motivation.  I'd like to try to explain the motivation here, and reintroduce the problem.

Consider binary sequences of length $2n$.  Let's say we put a marker in such a sequence as soon as we see a total of $n$ 0's or $n$ 1's, reading left to right.  For example, if $n=4$, then the sequence 00101011 would receive a marker thus:  001010|11.  Now write down the bits to the right of the marker.  In the case of our example, this would be 11.  Do this for every binary sequence of length $2n$.  We observe that we have written down 
$2n\binom{2n}{n}$ bits, half 0's and half 1's.  It is possible to prove this observation using binomial coefficient identities, but I wonder whether there is a simple bijective proof.  

The previous urn problem was an equivalent probabilistic formulation of the case $n=5$.