# Will two convex hulls overlap?

I ran into the following problem while working in neural nets.

Given natural numbers $b$ and $r$, uniformly randomly choose $b+r$ points within a unit square. Call the $b$ points the blue points and the $r$ points red points. What is the probability $p(b,r)$ that the convex hull of the blue points, $H_b$ , overlaps with $H_r$?

Partial answer : I can't immediately think of anything but a multi-fold brute-force integral for this. Intuitively, it seems to me (I could be incorrect) that $p(b,r)$ has to satisfy $\lim_{b\rightarrow \infty} p(b,r) = 1$ and likewise $p(b,r) \rightarrow 0$ as $b$ or $r$ approach 0. Also, given $b$ random points, we can compute the expected size of its convex hull according to this paper , though it's not clear to me how to use this.

I don't know how to connect these disparate hints at solution and would like suggestions.

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