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Take a random $n \times n$ nonnegative matrix $D$. What is the probability that it is an abstract distance matrix, i.e. satisfies $D_{xy}+D_{yz} \geq D_{xz}$ for all index triples $x,y,z$?

In other words: what is the probability that a nonnegative function on $V \times V$, where $V$ is some finite set, defines a finite metric space?

Take a random $n \times n$ nonnegative symmetric matrix $D$ with zero diagonal. What is the probability that it is an abstract distance matrix, i.e. satisfies $D_{xy}+D_{yz} \geq D_{xz}$ for all index triples $x,y,z$?

In other words: what is the probability that a nonnegative function on $V \times V$, where $V$ is some finite set, defines a finite metric space?