2010 AMC 12B Problem 18

This is a continuous (geometric) probability. Anchor the second jump from $P=(0,0)$ to $Q=(1,0),$ and let $\alpha,\beta$ be the directions of the first and third jumps, so the start is $A=(\cos\alpha,\sin\alpha)$ and the end is $B=(1+\cos\beta,\sin\beta).$ Taking $0\le\alpha\le\pi$ and $0\le\beta\le2\pi,$ the requirement $AB\le1$ holds exactly when $\alpha\le\beta\le\pi.$ In the $\alpha\beta$-rectangle of area $2\pi^2,$ the favorable region is a triangle of area $\tfrac{\pi^2}{2},$ so the probability is $\dfrac{\frac{\pi^2}{2}}{2\pi^2}=\dfrac14.$ Thus, the correct answer is C.