Robert J. Vanderbei, in his book "Linear Programming, Foundations and Extensions", Chapter 5 "Duality Theory", section Notes, states that
The solution to Exercise 5.13 suggests that a random Linear Programming problem is infeasible with probability $1/4$, unbounded with probability $1/4$, and has an optimal solution with probability $1/2$.
(There is an online version of Vanderbei's book; go to p.98 of the pdf file, which is p.82 of the book)
Now, if a random problem has an optimum with probability $1/2$, then the rest follows by symmetry. Unfortunately, the author does not provide any further details on what makes him think so, and neither does he say for which kind of probability distribution such a claim may be true. The solution to the mentioned exercise boils down to the fact that for an unfeasible problem (with $P=\emptyset$) we have
$$\max\{f(x)~|~x\in P\}=-\infty<\infty=\min\{f(x)~|~x\in P\}$$
which is somewhat counter-intuitive, and to the fact that a problem and its dual may both be infeasible. However, those facts do not help me at all to understand Vanderbei's line of reasoning.
Can some of you see why he reaches the above conclusion?