It is known [de la Vega & Karpinski 2002] that random instances of MAX-3-SAT on $n$ variables can be approximated up to fraction at least 8/9 w.h.p. tending to 1 as $n$ tends to infinity.
Should one expect this to be close to optimal? In fact, is it possible, that random MAX-k-SAT instances can be approximated within any desirable fraction w.h.p.? At least on the face of it - this should not contradict the PCP theorem, whose output SAT instances are very construction-specific (Dinur) and are thus sets with negligible probability in this respect.