  We determine the inner product on the Hilbert space of wavefunctions of the
universe by imposing the Hermiticity of the quantum Hamiltonian in the context
of the minisuperspace model. The corresponding quantum probability density
reproduces successfully the classical probability distribution in the $\hbar
\to 0$ limit, for closed universes filled with a perfect fluid of index $w$.
When $-1/3<w\le 1$, the wavefunction is normalizable and the quantum
probability density becomes vanishingly small at the big bang/big crunch
singularities, at least at the semi-classical level. Quantum expectation values
of physical geometrical quantities, which diverge classically at the
singularities, are shown to be finite.
