Consider a random walk in one dimension.

Consider a numberline with integers marked on it. A drunkard it at x=0 at t=0. At each time interval he jumps +1 or -1. i.e. at t=1 he can be at x=1 or x = -1

After t = n, let him be at x_n.

What is the probability that |x_n| =< a, where a is some +ve integer.

Any ideas to start on this?

Also I would later want to generalize it to include conditions like

1. x_n > a

2. x_n < b

and combine them using AND, OR.

So please advise. Thanks