Conditional probability of points in a plane

Question

I've ran into a problem with a project of mine and it boils down to the following example. Suppose I've a number of points dispersed in the region defined by $[0,100]\times[0,100]$ in $\mathbb{R}^2$. The region is allowed to "wrap around" on itself, so a point at coordinate $(0,57)$ is distance $1$ from a point at coordinate $(100,56)$. The number of points is immaterial, but for the sake of clarity let's say there are $1000$ points. Note that the points need not have integer coordinates. Each point is given a color, so there are $500$ blue points and $500$ red points. What is the probability that, given a blue point, there is a red point within distance $\varepsilon$ of the blue point?