  Quantum Mechanics (QM) is a quantum probability theory based on the density
matrix. The possibility of applying classical probability theory, which is
based on the probability distribution function(PDF), to describe quantum
systems is investigated in this work. In a sense this is also the question
about the possibility of a Hidden Variable Theory (HVT) of Quantum Mechanics.
Unlike Bell's inequality, which need to be checked experimentally, here HVT is
ruled out by theoretical consideration. The approach taken here is to construct
explicitly the most general HVT, which agrees with all results from experiments
on quantum systems (QS), and to check its validity and acceptability. Our list
of experimental facts of quantum objects, which all quantum theories are
required to respect, includes facts on repeat quantum measurement. We show that
it plays an essential role at showing that it is very unlikely that a classical
theory can successfully reproduce all QS facts, even for a single spin-1/2
object. We also examine and rule out Bell's HVT and Bohm's HVT based on the
same consideration.
