I thought of this question when I was walking aimlessly around my neighborhood. Here's my question:
My house is on an infinite lattice of points: say my house is at $(0,0)$. I start walking north (it doesn't matter what direction I start) and when I reach an intersection (at a lattice point) I choose to go left, right, or forward (relative to the direction I'm facing). What's the probability that I end up back at $(0,0)$, my house?
I have a feeling that the probability is $1$ because, very non-mathematically speaking, if I take an infinite amount of steps surely I will end up back at $(0,0)$ eventually, right? Also, I know of this problem in 1D, in which case I know the answer is $1$. I would prefer an answer that is intuitive.
It really depends also on the probabilities associated to which direction you travel in. If they are all equal, then for 1-d and 2-d such walks, you do come back to the origin. Surprisingly enough in 3-d, it is not possible, and the bird that flies out of the cage, never returns.. or some such thing!