A lost tourist arrives at a point with 3 roads. The road brings the tourist back to the same point after 1 hour of walk. The second road brings the tourist back to the same point after 6 hours of travel. The last road leads to the city after 2 hours of walk.
Assuming that the tourist chooses a road equally likely at all times and does not remember the past choices. What is the expected time until the tourist arrives to the city? Hint: use the law of total expectation to write an equation of the expected time.
My Attempt: I can assume that time must be at least 2 hours and $P[X=2] = 1/3$
By the hint I can write down: $E[X] = E[X|X=2]P[X=2] + E[X|X>2]P[X>2]$
But that's as far as I can go. I tried to calculate from the conditional expectation but I just can't think of a way to calculate it as I need the pmf.
Don't partition on the time it takes to walk to the city, rather, partition on the next road taken.
Let $\mathsf E(T)$ be the expected time taken to reach the city from the crossroad, and $\mathsf E(T\mid R{=}r)$ be the expected time when given that the tourist takes road $r$ (which may or may not loop back to the crossroad).
Thus the Law of Total Expectation you have been hinted to use is: $$\begin{align}\mathsf E(T)&=\mathsf P(R{=}1)~\mathsf E(T\mid R{=}1)+\mathsf P(R{=}2)~\mathsf E(T\mid R{=}2)+\mathsf P(R{=}3)~\mathsf E(T\mid R{=}3)\\[1ex]&=\tfrac 13(\mathsf E(T\mid R{=}1)+\mathsf E(T\mid R{=}2)+\mathsf E(T\mid R{=}3))\end{align}$$