Expected Value of a "composite" Random Variable
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 that will take the mouse back to the room after two days
- The second door leads to a tunnel that will take it back to the room after four days
- The third door will lead the mouse to freedom after one 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 disconsidered
- The mouse is 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!)