Relation via derivative of two angle-based probabilities for $n$ points on a circle. Suppose we sample $n$ points uniformly (and independently) on the unit circle in the sense that the probability that a point lies within some circular arc is proportional to its length. For this question, we'll choose the convention to measure angles from 0 to 1 (rather than from 0 to $2\pi$ when we use radians) to make the formulae come out nicer.
Then if we are given a fixed circular arc subtending an angle $0 \leq \alpha\leq \frac{1}{2}$ (the bounds chosen due to technicalities from reflex angles), we know that the probability that all $n$ points lie in this arc is $\alpha^n$. It is also a known result/elementary exercise that the probability that $n$ such points lie within some arc which subtends an angle of $\alpha$ is $n\alpha^{n-1}$.
Is there some probabilistic reason that these formulae are related by differentiation with respect to $\alpha$ ?
 A: I’m not sure whether this is what you’d call a probabilistic reason for this relationship, but I think it throws at least some light on why these two results should be the same.
The derivative of the probability that all points lie in a fixed arc is the rate of change of that probability as you expand the arc. The probability increases due to the possiblity that one of the $n$ points lies in the newly added segment and the $n-1$ others were already in the arc. There are $n$ ways to choose the point, the probability that it lies in the added segment of length $\mathrm d\alpha$ is just $\mathrm d\alpha$ (since the density is $1$), and the probability that the other $n-1$ points were already in the arc is $\alpha^{n-1}$, so the probability increase is $\mathrm dp=n\alpha^{n-1}\mathrm d\alpha$.
This is pretty much the same construction that leads to the result that the probability for all points to lie in some arc of length $\alpha$ is $n\alpha^{n-1}$: There are $n$ ways to choose the point that is farthest from its nearest neighbour in clockwise direction, and the probability that the other $n-1$ points are all within $\alpha$ of it in counterclockwise direction is $\alpha^{n-1}$.
