Consider a 3D volume $V$. $V$ contains all and only all the points $\langle x,y,z \rangle$ such that $0 \leq x,y \leq \sqrt{n}$ and $0 \leq z \leq 2\sqrt{n}$.
Consider a segment $A$ and a segment $B$ lying inside $V$. All the points of $A$ have z-coordinate greater than $0$ and less than $\sqrt{n}$; all the points $B$ have z-coordinate greater than $\sqrt{n}$ and less than $2\sqrt{n}$. The only additional constraint on $A$ and $B$ is that their length is $\sqrt{n}$.
I should evaluate the probability that a segment $C$ of length $\sqrt{n}$, lying inside $V$ and with an end point lying in the intersection between $V$ and the plane of points with $z=0$, intersects both $A$ and $B$.
The probability is roughly $1/n$ -- but what are the additional constraints required for $A$, $B$ and $C$?
