Do the lengths of all three segments necessarily have the same distribution?

Question

Let $A$ and $B$ be independent $U(0, 1)$ random variables. Divide $(0, 1)$ into three line segments, where $A$ and $B$ are the dividing points. Do the lengths of all three segments necessarily have the same distribution?

Answer 1

The answer is yes. The lengths of the three pieces are $L:=\min(A,B)$, $M:=|A-B|$, and $R:=1-\max(A,B)$. The distributions of these are found via the calculations (for $0<x<1$): $$ P(L>x)=P(\min(A,B)>x)=P(A>x,B>x)=P(A>x)P(B>x)=(1-x)^2 $$ $$ P(R>x) = P(1-\max(A,B)>x)=P(\max(A,B)<1-x)=P(A<1-x,B<1-x)=(1-x)^2 $$ and $$ P(M>x) = P(|A-B|>x) =(1-x)^2 $$ by noting that the region $\{|a-b|>x\}$ in the $(a,b)$-plane consists of two triangles whose union is a square of side length $(1-x)$.

Answer 2

The answer is positive, and one does not need any calculations. Let us work on the event $M = \{A<B\}$. Conditionally on $B$, $A$ is uniformly distributed on $[0,B]$. Therefore, $A$ and $B-A$ have the same conditional distribution given $B$, consequently, their unconditional distributions are the same. Similarly, $B-A$ and $1-B$ have the same distribution.

The same argument combined with induction proves that a similar result holds for any number of random variables.