Compute the Probability for Two Uniform Variables

Suppose $X$ and $Y$ are independent random variables, each uniformly distributed on $[0,1]$. Define:

• $L = \max(X,Y)$
• $S = \min(X,Y)$ What is the probability that the square of the larger value is greater than the smaller value, that is, $P(L^2 > S)$?

Overview: The problem checks your grasp of continuous probability, joint distributions, and order statistics for independent uniform variables, with attention to comparing transformations of the random variables.