# Probability that a random weight function on $K_n$ satisfies the triangle inequality

On a complete graph $K_n$, every edge is assigned a random real weight in $[0, 1]$. I am trying to calculate the probability that the weights satisfy the triangle inequality or even bounds on this probability. How about the discrete version where the weights are integers in $[0, k]$?

For $n=3$ the probability is $\frac12$. Empirically, for $n=4$ the probability seems to be between $0.141$ and $0.142$ so I doubt there is an easy expression for large $n$. –  Henry Jun 14 '12 at 23:15