# Maximum of two independent uniform random variables

Let $x$ and $y$ be uniformly distributed, independent random variables on $[0,1]$. What is the probability that the maximum between $x$ and $y$ is less than $1/2$ and greater than $1/3$?

• Draw a picture of $[0,1]$ and mark off the areas where $y \le x$ and $x \in [{1 \over 3}, {1 \over 2}]$. – copper.hat May 15 '15 at 15:42

The probability that both $x$ and $y$ are less than $1/2$ is $1/4$. The probability that both $x$ and $y$ are less than $1/3$ is $1/9$. We want the former to be true, but not the latter, so the answer is $1/4 - 1/9 = 5/36$.