Expected length of time it takes the Golem to reach safety? > Iron Golem is an action figure in Minecraft. Iron Golems are loyal to villagers, and will defend them from Hostile Mobs and siege attacks. Suppose that in a particular setting, an Iron Golem is trapped by Mobs in a mine containing 3 doors. The first door leads to a tunnel that will take him to safety after 3 hours of travel. The second door leads to a tunnel that will return him to the mine after 5 hours of travel. The third door leads to a tunnel that will return him to the mine after 7 hours of travel. If we assume that the Iron Golem is at all times equally likely to choose any one of the doors, what is the expected length of time until he reaches safety? This question feels like I should be approximating it to either the hyper-geometric/negative binomial models and maybe having more than one distribution function. But I'm totally stuck after that thought.

Since two of the three choices lead you right back to the initial state, you can compute the expected escape time fairly easily. Letting $E$ be the expected escape time, we have $$ E=\frac13\cdot3+\frac13\cdot(5+E)+\frac13\cdot (7+E), $$ therefore $E=15$.