Random rectangle hitting the middle row of a grid

In a $2023\times2023$ grid of unit squares, all squares in the middle row (row $1012$) are shaded. A rectangle made of whole grid squares is chosen uniformly at random among all such rectangles. What is the probability it includes at least one shaded square?

$\tfrac{1012}{2023}$. Rectangles are pairs of horizontal and pairs of vertical grid lines. It misses the shaded row only if both horizontal lines lie among the $1012$ above it or the $1012$ below it, probability $\tfrac{2\binom{1012}{2}}{\binom{2024}{2}}=\tfrac{1011}{2023}$; the complement is $\tfrac{1012}{2023}$.