Maximum function on I.I.D random variables

Question

Let $X_1$ , $X_2$ , $X_3$ , $X_4$ be i.i.d random variables U [$0$,$1$]

How do I find P [ $X_3$ < $X_2$ < max ($X_1$,$X_4$) ] ?

I know that this problem can be solved using the exchangeable property of i.i.d random variables. However, I tried solving this problem using the conventional integration/required region way, but my answer was coming wrong. How do I solve this problem using integration method.

For example, I tried doing this: P [ $X_3$ < $X_2$ < max ($X_1$,$X_4$) ]

= P [ $X_3$ < $X_2$ < $X_1$ > $X_4$) ] + P [ $X_3$ < $X_2$ < $X_4$ > $X_1$ ]

= 2 $\int_{0}^{x_2} \int_{x_3}^{x_1}\int_{x_2}^{1}\int_{0}^{x_1} \,dx_4dx_1dx_2dx_3 $

Answer 1

The limits of integration are not right. The limits of an integral cannot depend on dummy variables that are integrated away in inner integrals.

$$P(X_3 < X_2 < X_1 > X_4) = \int_0^1 \int_{x_3}^1 \int_{x_2}^1 \int_0^{x_1} dx_4\, dx_1 \, dx_2 \, dx_3$$