All random points on a circle in one half

Pick $N$ points independently and uniformly on the circumference of a circle. What is the probability that some semicircle contains all of them?

$N/2^{N-1}$. For each point, the event that all the others lie in the semicircle running clockwise from it has probability $2^{-(N-1)}$. These $N$ events are disjoint (at most one point can be the clockwise-first point of such a semicircle) and their union is the target event, so the probability is $N\cdot2^{1-N}$.