AMC 12 2023b Problem 20

Cyrus the frog jumps $2$ units in a direction, then $2$ more in another direction. What is the probability that he lands less than $1$ unit away from his starting position?

\textbf{(A)}\frac{1}{6} \textbf{(B)}\frac{1}{5} \textbf{(C)}\frac{\sqrt{3}}{8} \textbf{(D)}\frac{\arctan \frac{1}{2}}{\pi} \textbf{(E)}~\frac{2\arcsin \frac{1}{4}}{\pi}