Four random points inside a quarter-circle arc

Choose $N$ points independently and uniformly on the circumference of a circle. For an angle $0\le x\le\pi$, what is the probability that some arc of angle $x$ radians contains all of them? Evaluate it for $N=4$ and $x=\pi/2$.

$N\left(\tfrac{x}{2\pi}\right)^{N-1}$, which is $1/16$ here. For each point, consider the event that the others all lie within angle $x$ clockwise from it; when $x\le\pi$ at most one point can be such a leader, so the probabilities add: $N(x/2\pi)^{N-1}=4\cdot(1/4)^3$.