There are three doors in a Maze. Choosing each door equal probability. If we choose the $1^{st}$ door, then after 5 minutes, we can go out of the Maze. If we choose the $2^{nd}$ door, then after 10 minutes, we will come back in the same door. If we choose the $ 3^{rd}$, then after 7 minutes, come back in the same door. Nothing remember after return to the same door.
I tried to solve this equation, but for the $2^{nd}$ & $ 3^{rd}$ door, I don't understand how to solve. $$ E = \frac{1}{3}(5) + \frac{1}{3}(?) + \frac{1}{3}(?) $$
After $10$ or $7$ minutes you are back where you started, which means the expected time is $$. The first question mark should be $10+E$ and the second should be $7+E$