[Math] Independent Random variables, identically distributed

Question

Let $X,Y,X$ be random variables defined on the same probability space $(\Omega,F,P)$. Suppose that $X,Y,Z$ are independent and identically distributed and the common distribution is continuous. Prove that
$\displaystyle P\{X<Y<Z\}=\frac{1}{3!}.$

My try: well I've done this in a particular case, and as the problem says, it's true when
$X, Y, Z$ have exponential distributions with parameter $\lambda$. There it is easy because the problem reduces to calculating the integral
$\displaystyle \int_{0}^{\infty}\int_{x}^{\infty}\int_{y}^{\infty}\lambda^{3}e^{-\lambda(x+y+z)}dz dy dx.$ But I don't know how to do it in general. Any suggestions? Thanks beforehand.

Answer 1

Hint: ${\rm P}(X < Y < Z) = {\rm P}(X < Z < Y) = \ldots$.