The problem could be stated as follows : we have some random walker in an unbounded 1-dimensional lattice, such that there is a 50% chance the walker doesn't move at all, a 25 % chance the walker moves to the left, and 25% chance the walker moves to the right. What is the probability of the walker ending up at some point in the lattice in $N$ steps?
If we now denote the position of the walker as an integer i.e. $1$ would refer moving one site in the lattice to the right, and $-1$ would refer to moving to the left. Then what sort of a distribution would describe the probability that the in $N$ steps the walker would end up at some specific point on the lattice? My intuition says that I am looking for sums of the terms in an $N$-tuple that add up to the point in the site. For instance the tuple described by $(1,1,-1,0,...,0)$ would put the walker at $1$ for the end point.