Two cuts on opposite halves farther apart than a third

Noelle will cut a fish of length $L$ at two points. One point is uniform on the left half and the other, independently, is uniform on the right half. What is the probability that the cuts are more than $L/3$ apart?

$7/9$. With $X\sim U[0,L/2]$ and $Y\sim U[L/2,L]$, the bad region $Y-X\le L/3$ is a right triangle with legs $L/3$, area $L^2/18$, out of total area $L^2/4$. So the answer is $1-\tfrac29=\tfrac79$.