Question
Jeff is trapped in a room with $5$ indistinguishable doors. Behind $2$ of the doors are paths to freedom; one path is $2$ miles long, and the other path is $5$ miles long. Behind $3$ of the doors are paths that magically lead back to the room; one path is $1$ miles long, another path is $3$ miles long, and the final path is $4$ mile long. $$$$ If Jeff returns to the room, the order of the doors are scrambled and the $5$ doors are once again indistinguishable. What is the expected number of miles Jeff will travel before reaching freedom?