How can I calculate the probability that one random variable is bigger than a second one? I have five random variables which are independent and each one of them has a continuous uniform distribution on the interval $ [0,2]:$
$$X_i = \operatorname{Uniform}[0,2].$$
I want to calculate the probability $$\Pr(\min(X_1, X_2, X_3)\gt \max(X_4, X_5)).$$
I'm aware there is combinatorial solution, but I'm trying to solve this problem using coordinates with $X$ as the minimum and $Y$ as the maximum, but I don't know how to sketch the function and calculate the function space in order to know the probability.

Solution using automated computer algebra systems:
Let $(X_1, ..., X_5)$ have joint pdf $f(x_1,..., x_5)$:

Then:

... where I am using the Prob function from the mathStatica package for Mathematica.
Always nice to check work and existence of exact solutions.
