2003 AMC 12B Problem 25

A chord is shorter than the radius exactly when its arc measures less than $60^\circ$

All three pairwise chords are short precisely when the three points lie within some $60^\circ$ arc

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.