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? $\textbf{(A)}~\frac{1}{6}\qquad$ $\textbf{(B)}~\frac{1}{5}\qquad$ $\textbf{(C)}~\frac{\sqrt{3}}{8}\qquad$ $\textbf{(D)}~\frac{\arctan \frac{1}{2}}{\pi}\qquad$ $\textbf{(E)}~\frac{2\arcsin \frac{1}{4}}{\pi}$
Solution
Place the first jump along the $x$-axis, landing at $(2,0)$; let the second jump be at angle $\alpha$ from the $x$-axis, landing at $(2+2\cos\alpha,2\sin\alpha)$.
Squared distance from origin: $(2+2\cos\alpha)^2+(2\sin\alpha)^2=8+8\cos\alpha$. We need this $<1$: $\cos\alpha<-\frac78$.
This holds for $\alpha$ in an arc of measure $2\pi-2\arccos\left(-\frac78\right)=2\arccos\frac78$ (using $\arccos(-x)=\pi-\arccos x$), out of total $2\pi$: probability $=\dfrac{\arccos(7/8)}\pi$.
Since $\arccos\frac78=2\arcsin\frac14$ (as $\cos(2\arcsin\frac14)=1-2\cdot\frac1{16}=\frac78$), the probability is $\dfrac{2\arcsin(1/4)}\pi$, answer (E).
Cyrus the frog jumps $2$ units in a direction, then $2$ more in another direction. What is the probability that he lands less than $\sqrt2$ units from his starting position?
