Past exam of the mathematics course of the University of Cambridge / 2002 / ia / Paper 2 / 11F / c / Solution

Three positive lengths form a nondegenerate triangle precisely when each is less than the sum of the other two. Necessity is the triangle inequality. For sufficiency, two circles centered at the ends of one side intersect off that side whenever the remaining lengths satisfy these strict inequalities, giving the third vertex.

Failure thus occurs when $X\ge Y+Z$, $Y\ge X+Z$, or $Z\ge X+Y$. Two such inequalities cannot hold together for strictly positive lengths. Zero lengths and equality cases have probability zero because the variables have continuous probability density functions. By symmetry each failure event has the probability $1/6$ just calculated. Hence the probability of forming a triangle is