No. The laws of thermodynamics and statistical physics^1 applied only to large collections of particles, in so-called thermodynamic limit, where the number of particles $N$ and the volume $V$ of the system go to infinity, but some parameters, like density $n=N/V$ are kept constant. Typically, one takes Avogadro number of particles $N_A\approx 10^{24}$ as the case where the corrections to the quantities of interest (aka fluctuations) are negligibly small. However with appropriate care one can apply thermodynamic laws to much smaller systems, keeping in mind that the fluctuations decrease as $\propto \frac{1}{\sqrt{N}}$.
Remark:
Not sure if this is behind the question: randomness inherent in quantum mechanics should not be confused with the statistical (but classical) randomness in statistical physics. Obviously, both are combined when we deal with large collections of quantum objects.
Update in response to comments
As pointed above, statistical mechanics deals with large collections of particles, and the statistical averaging is understood as averaging over all the possible configurations of particles (e.g., their positions, momenta, spin, etc.)
In traditional interpretations of quantum mechanics (like the Copenhagen interpretation) averaging is understood as averaging over many identical experiments on identical particles/systems prepared in the same state. Mathematically, it is also statistical averaging, but over a different ensemble. Such an averaging is necessitate by the fact that the measurement collapses the wave function (unless it is in the eigenstate of the measurement operator), and thus cannot be repeated on the same system. Note however, that this ensemble is meaningful even for a single particle (where we cannot talk about thermodynamic properties.)
States of a collection of quantum particles are described by multiparticle wave functions, which often cannot be factorized into the product of wave functions of single particles (due to interactions, and due to the indistinguishability of particles, which imposes (anti)symmetry on the many-particle wave function.) Different configurations of the many-particle system then refer to different eigenstates (wave functions) of the many-particle system, and statistical averaging implies averaging over these states (i.e., "averaging of wave functions".) As the result, in *quantum statistical physics one has to resort to using the density matrix - a construction more general than wave function.
^1 Statistical physics is probably more correct term here, since we talk about microscopic description rather than phenomenology)
