2003 AMC 12B Problem 25

A chord has length less than the radius exactly when the arc it subtends is less than $60^\circ,$ since a chord of a $60^\circ$ arc equals the radius. All three pairwise chords are shorter than the radius precisely when the three points all lie within some arc of $60^\circ.$ For any successful configuration, exactly one of the three points is the counterclockwise endpoint of such a containing arc (apart from probability-zero boundary cases). Choose that endpoint in $3$ ways; each of the other two points independently has probability $\dfrac{1}{6}$ of lying in the next $60^\circ.$ Hence the probability is $$3\left(\frac{1}{6}\right)^2 = \frac{1}{12}.$$ Thus, the correct answer is D.