Rat in a Maze

A rat is in a maze, facing two doors. At each attempt, it chooses:

• the first door with probability $p$, and
• the second door with probability $1 - p$.

The consequences are:

• If it chooses the first door, it immediately returns to its starting point after one minute.
• If it chooses the second door, it moves to an intermediate point in one minute, then:
  • it turns back with probability $q$, which takes another one minute, or
  • it leaves the maze permanently with probability $1 - q$, after one minute.

All of the rat's choices are made independently of each other.

Let $T$ denote the total time spent by the rat in the maze.

Questions:

1. Determine the expectation $\mathbb{E}[T]$.
2. Determine the probability distribution of $T$.