What is the chance that two freely rotated ends of a broken stick will intersect? Given three line segments $L1$, $L2$, and $L3$ of length $1$, give them each a random orientation chosen independently and uniformly among angles, then connect $L1$ to $L2$ and $L2$ to $L3$. What's the probability that $L1 $ intersects  $L3$
As I understand it, the largest angle at which we will intersect $L1$ and $L3$ is 60 degrees, and the smallest will be equal to 0? but most likely I don't understand which angle will be the smallest here. And as I understand it, it is better to solve this graphically

I'm going to make some assumptions about the problem so that the problem makes sense to me.
Assume the line segments have been oriented so that L1 and L2 have one endpoint at the origin and that L3 is connected to the other end of L2 in a particular way defined below (there are other ways to define the ways of connecting, but this is one reasonable way).
Without loss of generality, assume:

*

*L2 has endpoints $(0,0)$ and $(1,0)$

*L1 has endpoints $(0,0)$ and $(\cos{\theta_1}, \sin{\theta_1})$

*L3 has endpoints $(1,0)$ and  $(1-\cos{\theta_2}, \sin{\theta_2})$
where $\theta_1$ and $\theta_2$ are independent and identically distributed uniform on $[0,2\pi]$.
Now, for each fixed $\theta_1$, find the set of $\theta_2$ that will result in L1 intersecting L2. This is tricky.
To make it easier, just consider the case where $\theta_1<\pi$. By symmetry, the probability that L1 intersects L3 will be twice the probability that L1 intersects L3 and $\theta_1<\pi$.
For $\theta_1$ between 0 and $\pi/3$, the $\theta_2$ are those in $[\pi/2+\theta_1/2,\pi]$. For $\theta_1$ between $\pi/2$ and $\pi$, the $\theta_2$ are those in $[2\theta_1,\pi]$. For other $\theta_1<\pi$, the lines segments do not intersect.
Therefore, the probability is
$$2\left(\int_{0}^{\pi/3}{\frac{\pi/2-\theta_1/2}{4\pi^2}}d\theta_1+\int_{\pi/3}^{\pi/2}{\frac{\pi-2\theta_1}{4\pi^2}}d\theta_1\right)=\frac{1}{12}.$$
There are other ways of connecting the line segments. Some of them result in problems with trivial solutions or solutions that can be derived from the solution above (for example if all three line segments are connected at the same point). But, a harder problem is when I assume the lines are connected not at the endpoints, but at a random point in the interior of the line segments. I can only approximate the solution for this case by simulation.
