Probability of points on a Line

Question

Three points $X_1,X_2,X_3$ are selected at random on a line of length $L$. What is the probability that $X_2$ lies between $X_1$ $and$ $X_3$?

I know that all three are equally likely to be in the middle, but I don’t know how that is so, I’m looking for a way to come to that answer.

Answer 1

Clearly $1/3$ due to permutation symmetry.

Answer 2

Assuming that $X_{i}$ is coming from a continuous uniform distribution on the interval $(0,L)$ then $P(X_{i}=a)=0$, for any $a$ on the real line. Note that there are many ways to construct an interval of length $L$, such as $(-\frac{L}{2},\frac{L}{2}$), but I will assume in this case, the interval is $(0,L)$. Then your question is asking to find:

$P(X_{3}<X_{2}<X_{1})$

=$\int_{x_{3}=0}^{L} \int_{x_{3}}^{L}\int_{x_{2}}^{L} \frac{1}{L^{3}}dx_{1}dx_{2}dx_{3}$