Who keeps the ball in a five-person circle

Five people, including you, sit in a circle, and you hold a ball at the start. Each second, whoever holds the ball picks one of three actions uniformly at random: pass it to the neighbour on the left, pass it to the neighbour on the right, or keep it. Play ends as soon as a holder decides to keep it. Find the chance that the game finishes with the ball in your hands.

$5/11$. By symmetry only three states matter: you hold it ($x$), a neighbour holds it ($y$), or one of the two far players holds it ($z$). Then $x=\tfrac13+\tfrac23y$, $y=\tfrac13x+\tfrac13z$, $z=\tfrac13y+\tfrac13z$. These give $z=y/2$, $y=2x/5$ and finally $x=5/11$.