Probability of two points being a certain distance apart on a circle

Question

Is the probability of two points being a certain distance $k$ apart on a circle of length $m$, with $0\le k<\frac{1}{2}m $, always the same for any $k$?

Answer 1

The short answer is "yes," in the sense that if the two points were determined by a probability distributed uniformly and independently (and continuously) along the circumference, then the distance is a random real-valued variable uniformly distributed over the interval $[0,\frac12 m].$

Answer 2

Short answer is "yes".

Probability that the distance is exactly $k$ is zero. But I guess your question was about probability of a distance to be "around" $k$, somewhere between $k$ and $k$ + $dx$. If $k$ is valid distance, this probability depends on $dx$ only, not on $k$ itself.

You choose the first point randomly, and after that suitable region for the second point consists of two arches, each having a length $dx$. Probability that the second point would get into this region is $2 * dx / m$. It does not depend on on position of the first point and does not depend on $k$.