TITLE: let x and y be uniformly distributed independent random variables on [0 ,1].the probability that the distance between x and y is less than 1/2 is?
QUESTION [5 upvotes]: I have a question about probability:
let x and y be uniformly distributed independent random variables on [0 ,1].the probability that the distance between x and y is less than 1/2 is?
can someone give some hints to compute it ?

REPLY [3 votes]: Consider $x$ and $y$ as the axes on the graph. Then all values of $(x,y)$ will lie on the square $[0,1] \times [0,1]$. Now for any fixed value of $x \in [0, 1]$, try to find the region of the square that has values of $y$ such that $|y - x | < 1/2$.
Observe that this region is a band around the line $y = x$ and using simple geometry you can calculate its area to be $3/4$.