This is the typical drunk problem wherein the person is confined to moving either to the North, South, East, or West but never diagonally with just one step. A step has a length $L$. What is the probability that the drunk will never leave a circle of radius $2L$ after $N$ steps?
Obviously, the probability is zero for $N=1$ and $N=2$. For $N=3$, I got it to be $3/4$ although I am not sure whether this is correct. For $N>3$, I am just lost.