The problem is indeed choosing uniformly randomly two point $x,y$ on the interval $[0,1]$ such that the length of each sub-interval is less than $\frac{1}{2}$. This is equivalent to probability that two points chosen uniformly randomly on the interval $[0,1]$ fall into the interval $[0,\frac{1}{2}]$ which is $\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}$.

To see this, it is clear that one point, say $x$, should be in $[0,\frac{1}{2}]$ and another one, say $y$, in $[\frac{1}{2},1]$. Now translate $y$ backward by $\frac{1}{2}$ to the point $y'$. Now $y'<x$ for the desired event, i.e. for the case of having the length of each sub-interval less than $\frac{1}{2}$. This means each desired event is equivalent to having two points in $[0,\frac{1}{2}]$ and translate the first one by $\frac{1}{2}$.

The problem is indeed choosing uniformly randomly two points $x,y$ on the interval $[0,1]$ such that the length of each sub-interval is less than $\frac{1}{2}$. This is equivalent to probability that two points chosen uniformly randomly on the interval $[0,1]$ fall into the interval $[0,\frac{1}{2}]$ which is $\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}$.

To see this, it is clear that one point, say $x$, should be in $[0,\frac{1}{2}]$ and another one, say $y$, in $[\frac{1}{2},1]$. Now translate $y$ backward by $\frac{1}{2}$ to the point $y'$. Now $y'<x$ for the desired event, i.e., for the case of having the length of each sub-interval less than $\frac{1}{2}$. This means each desired event is equivalent to having two points in $[0,\frac{1}{2}]$ and translate the first one by $\frac12$.