We consider the SO(3) Witten-Reshetikhin-Turaev quantum invariants of random
3-manifolds. When the level r is prime, we show that the asymptotic
distribution of the absolute value of these invariants is given by the standard
Rayleigh distribution and independent of the choice of level. Hence the
probability that the quantum invariant certifies the Heegaard genus of a random
3-manifold of a fixed Heegaard genus g is positive but very small, less than
10^-7 except when g < 4. We also examine random surface bundles over the circle
and find the same distribution for quantum invariants there.