Question

Problem 2. You find yourself trapped underground in the mines of Moria, and you come to a room that contains three doors. One door leads to a tunnel that will lead you out of the mine after 9 hours of walking. Another door leads you back to the same room after 6 hours of walking. The third door leads to a fork in the path such that half the time you'll exit the mine and half the time you return back to the same 3-door room as before. Both forks take 12 hours of time. Because of the darkness and your exhaustion and dehydration, each time you return to this room you will have completely forgotten what door you may have chosen previously. Thus assume that each time you visit this room, you always select one of the three doors completely at random, independently of any previous choices With this assumption, what is the expected time that you will spend in the mine?

Answer 1