2013 AMC 12A Problem 24

Inscribe the $12$-gon in a unit circle. The segment lengths are $d_k = 2\sin(15k^\circ)$ for $1 \le k \le 6,$ with $12$ segments of each length $d_1, \ldots, d_5$ and $6$ of length $d_6.$ Comparing sums, the forbidden index triples $(a, b, c)$ with $d_a \le d_b \le d_c$ and $d_c \ge d_a + d_b$ are $$\begin{gathered} (1,1,3),(1,1,4),(1,1,5), \\ (1,1,6),(1,2,4),(1,2,5), \\ (1,2,6),(1,3,5),(1,3,6), \\ (2,2,6). \end{gathered}$$ The first three triples ending in $3,4,5$ contribute $3\binom{12}{2}12;$ the two repeated-length triples ending in $6$ contribute $2\binom{12}{2}6;$ the three triples of distinct non-diameter lengths contribute $3\cdot12^3;$ and the two remaining diameter triples contribute $2\cdot12^2\cdot6.$ Dividing their sum by $\binom{66}{3}$ gives failure probability $\frac{63}{286},$ so the answer is $1-\frac{63}{286}=\frac{223}{286}.$ Thus, the correct answer is E.