Absolute value distributions

Question

I have two iid random variables $X$ and $Y$, with both being Uniform on $[0,1]$. I want to find the probability that the distance between an arbitrary $x$ and $y$ is greater than 0.5. I tried constructing a third random variable, $Z=|X-Y|$, but I have no idea how to go about forming the distribution for $Z$. Is this the right way to think about this problem?

Answer 1

Note that $(X,Y)$ is uniformly distributed on the unit square $[0,1] \times [0,1]$. This means that the probability that $(X,Y)$ lies in some region of the square is equal to the area of the region.

For your problem, draw the region of the unit square consisting of points $(x,y)$ such that $|x-y| > 0.5$, and find its area.