If 6 numbers are chosen at random, uniformly and independently, from the interval [0,1], what is the probability that they are the lengths of the edges of a tetrahedron?

I wrote some code and simulate this probability and I suppose the answer is 1/3.

As for 2-dimensional question it is trivial to find that: If 3 numbers are chosen at random, uniformly and independently, from the interval [0,1], the probability that they are the lengths of the sides of a triangle is 1/2.

3-dimensional case seems to be very difficult to prove.

Moreover is this true (n-dimensional case):

If n(n+1)/2 numbers are chosen at random, uniformly and independently, from the interval [0,1], what is the probability that they are the lengths of the edges (1-faces) of a n-dimensional simplex?

Inspired by the 2 and 3-dimensional cases I suspect the answer is 1/n but this one seems to be a real chestnut.