= Embedded Markov chain at level exits
Suppose motion within each level ends on hitting one of two boundaries, and each boundary transition resets the process to a fixed entry state of an adjacent level. Observe only the successive level exits. The resulting level process is a <Markov chain>, with transition <probabilities> equal to the within-level boundary-hitting <probabilities>. The fixed-entry condition matters: without it, retaining only the level may discard information needed for the next transition law.
For a <simple symmetric random walk> on local rooms $0,\ldots,r$, starting each excursion at room $1$, the <gambler's ruin> <probability> of reaching $r$ first is $1/r$. Thus the level chain moves up with <probability> $1/r$ and down with <probability> $(r-1)/r$. Independent future decisions, or more generally the <Strong Markov property>, justify restarting this calculation after each exit.
For two active levels between absorbing levels $0$ and $3$, starting on level $1$, put $p=1/r$ and $q=1-p$. The success <probabilities> satisfy $u=pv$ and $v=p+qu$, hence
$$
u=\frac{p^2}{1-pq}.
$$
For $r=4$, the result is $1/13$. This compression illustrates how unbiased microscopic choices can produce a biased coarse <random walk> when entry points are close to one boundary.
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