2023 AMC 10B Problem 19

Sonya the frog chooses a point uniformly at random lying within the square [0, 6] \times [0, 6] in the coordinate plane and hops to that point. She then chooses a distance uniformly at random from [0, 1] and a direction uniformly at random from \{\text{north}, \text{south}, \text{east}, \text{west}\}. All her choices are independent. She now hops the distance in the chosen direction. What is the probability that she lands outside the square?

\textbf{(A)} \dfrac{1}{6} \qquad \textbf{(B)} \dfrac{1}{12} \qquad \textbf{(C)} \dfrac{1}{4} \qquad \textbf{(D)} \dfrac{1}{10} \qquad \textbf{(E)} \dfrac{1}{9}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.