# 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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If you can find the expected area of the convex hull of $b$ blue points (call it $E[A]$), then the rest is easy. The probability that one red point, chosen uniformly and independently of the blue points, will lie in the convex hull is then again $E[A]$ (to see this, condition on the location of the blue points). So the probability that the convex hulls overlap is just the probability that some red point lies inside the convex hull of the blue points, which by independence is $1-(1-E[A])^r$.