Escape distance through five shuffled doors

A man is locked in a room with $5$ identical-looking doors. Two doors lead out: one by a $2$-mile route and one by a $5$-mile route. The other three lead along routes of $1$, $3$ and $4$ miles that bring him straight back to the room. Each time he is in the room he picks a door uniformly at random, since the doors are reshuffled after every return. What is the expected total distance he walks before escaping?

$\tfrac{15}{2}$ miles. Condition on the first door: with $E$ the expected distance, $E=\tfrac15\big[(1+E)+(3+E)+(4+E)+2+5\big]$, i.e. $E=\tfrac35E+3$. Solving gives $E=7.5$.