Locked in a dungeon, you are faced with five doors. One of them leads to freedom. The other four will lead you back to the starting room disoriented and confused, so that you will not remember which of the doors you have already tried and have to start again.
How many attempts do you expect to make on the average (statistically) before making it out?
(In reply to re: Simpler Solution
N (Average number of doors we expect to go through before leaving) = 1 (try one of them...) + 0.8 * N (... and there's an 80% chance you're back in the room again with no better knowledge after trying that one, so the Number of doors you'll expect to have to go through if that happens is N again)
It does work and, as mentioned, it is much simpler.