Let's say there is a spider and an ant in a square box. The ant stays in the center of the box. The spider walks a random distance, in a random direction. What are the chances that he will meet the ant?
If you're talking about two geometric points representing the spider and the ant, the probability is 0. If one or both are modeled to occupy a measurable space, the probability is 1 given a finite space and no time limit.
The problem doesn't sound well-defined. How many possible directions are there for the spider to walk in? How many possible distances? (If you want to keep things simple, pick a finite number.) Where does the spider start? What is the shape of the path that the spider walks in? I can't agree with SW VandeCarr because, if the spider crawls in a straight line along the inside of the box, it's going to stay in one plane, and if the ant isn't in that plane, the spider can walk forever and never meet the ant.
I understood that by saying the spider walks in a random direction for a random distance, the OP was attempting to describe Brownian type motion in a finite space with no time limit. Otherwise the question makes no sense.