2016 AMC 12A Problem 23

The ordered triples $(x,y,z)$ fill the unit cube of volume $1.$ They fail to form a triangle exactly when one value is at least the sum of the other two. The region $z\ge x+y$ is a tetrahedron with vertices $(0,0,0),(0,0,1),(0,1,1),(1,0,1)$ of volume $\frac16.$ The analogous regions $x\ge y+z$ and $y\ge x+z$ also have volume $\frac16$ and have disjoint interiors. So the failure probability is $3\cdot\frac16=\frac12,$ and the triangle probability is $1-\frac12=\frac12.$ Thus, the correct answer is C.