The Largest Piece of Circumference We add $n$ random lines to the unit disc. We do so by adding two points on the disc, transcribing a line between them and extending that line to the boundary of the disc, namely the unit circle. Each line is chosen independently and we do this $n$ times. 
After $n$ random lines have been added, what is the probability that the largest arc on the unit circle between two points is of length $\pi/2$ or longer? To be clear, the arc itself must not have any points on it.
 A: The distribution of the maximal distance between a pair of random points on the circle is known - when you scale it by $n/\log n$ you get a Gumbel distribution with scale 1, location 1., see, e.g., 
Schlemm, Eckhard, Limiting distribution of the maximal distance between random points on a circle: a moments approach, Stat. Probab. Lett. 92, 132-136 (2014). ZBL1294.60045.
so it is quite obvious the the probability of maximal distance being $O(1)$ goes to zero exponentially fast (it is obvious that it goes to zero about as fast as $(3/4)^{2n}$, in fact.
A: To re-ask Pietro Majer's question, what do you mean by a "piece of circumference"?

          


          

$20$ random points, $n=10$ chords.


If you mean the largest section of the circumference containing no chord points,
then the chords play no role: the question could be posed just in terms of
$n$ (or $2n$) points, rather than $n$ "lines."
On the other hand, every chord of the circle has half the circumference to one
side or the other.

Added.
I think James Smith's interpretation of the question makes the most sense:

          


          

$10$ random points inside the circle, determining $n=5$ lines. Largest arc: $94^\circ$.


