Hi everybody! I'm presented with the following problem:

Consider a mouse (or rabbit, or other dumb aninal of your choice) that is inside a room (or cell) with THREE doors. The doors behave as follows:

The first door leads to a tunnel thatwill take the mouse back to the roomaftertwo daysThe second door leads to a tunnel thatwill take it back to the roomafterfour daysThe third door willlead the mouse to freedomafterone day

Other assumptions:

Every time the mouse is in the room, it chooses one door randomly, with equal probability.The time the mouse spends choosing a door can be disconsideredThe mouseis dumb enough to choose the same door more than one time. i.e., it can theoretically stay in the room forever if it never chooses the right door.

Given this situation, we ask,how long does it probably take until the mouse REACHES freedom?

Now my thought process: I believe I have to define a random varibale $\displaystyle T$ that represents the time the mouse stays not in freedom. And then the expected value of $\displaystyle T$ is the answer to the problem.

I believe I have the correct 'outline' to solve the problem, but I'm struggling with the individual steps. In particular, I couldn't manage to define the $\displaystyle T$ variable well, and then find its PDF to calculate the expected value.

So, any tips on how to start? (sorry for the verbose post!)