Probability that a stick randomly broken in five places can form a tetrahedron

and software related to it: an(other) attempt to
solve the problem numerically.
With 5 random cuts there exist 30 possible configurations.
Criterion: all triangle inequalities must be fulfilled and
Cayley-Menger determinant positive. At least one of the 30
possible tetrahedra should exist. 1,000,000 attempts, with
different seeds. Then some numerical approximations of the
probability that a tetrahedron can be formed are:
0.037177 0.037425 0.037722 0.037395 0.037396 0.037433
0.038100 0.037924 0.038292 0.037595 0.037858 0.037938
0.037298 0.037912 0.037691 0.038217 0.038116 0.037865

Much simpler problem in 2-D:

If you break a stick at two points chosen uniformly, the probability the three

resulting sticks form a triangle is 1/4. Is there a nice proof of this?