A particle moves $N$ steps as detailed here: in every steps it is to turn north, south, east or west, one unit at a time, with unchanging probability and with no dependence on any recent step.(That creates a uniform sample space, $|\Omega|=4^N$). What is the probability that it ends in its starting point?
I know I have to find an event A, and combinatorially compute $|A|$. My biggest problem is: for every single step I make north I walk one south(respectively) and do the same with east and west? Because if so, I think I can solve it on my own. This kind of question is confusing me, though. I could really use your observation.