Points on a semi-circle

Suppose you choose $N$ points uniformly randomly on a circle. What is the probability that all the points lie on a semi-circle.

Random walk on a circle

Let $x_1, x_2, ..., x_n$ be n points (in that order) on the circumference of a circle. Dana starts at the point $x_1$ and walks to one of the two neighboring points with probability $1/2$ for each. Dana continues to walk in this way, always moving from the present point to one of the two neighboring points with probability 1/2 for each.  $P_i$ is the probability that the point $x_i$ is the last of the n points to be visited for the first time.

1. Show that $P_i$ is the same for all i.

2. Find  $P_i$