Expected Distance to Escape a Room
You are stuck in a room with 4 doors. Two doors lead to freedom: one down a path of length 5 miles, the other 7 miles. The other two doors loop you right back to this same room -- one after 2 miles, the other after 4 miles. At every step you pick a door uniformly at random.
What is the expected total distance you travel before reaching freedom?
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