Long-run share of time a toddler spends at the couch

A toddler practises walking along a straight line starting from a couch. Each time step, it moves one step away from the couch with probability $0.2$, stays put with probability $0.5$, and moves one step back toward the couch with probability $0.3$. It cannot go behind the couch, so when it is at the couch it moves away with probability $0.2$ and otherwise stays there. Over a very long observation period, what fraction of the time is the toddler at the couch?

$1/3$. This is a birth–death chain; detailed balance gives $0.2\,\pi_n=0.3\,\pi_{n+1}$, so $\pi_n=\pi_0(2/3)^n$. Summing the geometric series, $\pi_0\cdot 3=1$, so $\pi_0=\tfrac13$.