  We discuss the classical statistics of isolated subsystems. Only a small part
of the information contained in the classical probability distribution for the
subsystem and its environment is available for the description of the isolated
subsystem. The "coarse graining of the information" to micro-states implies
probabilistic observables. For two-level probabilistic observables only a
probability for finding the values one or minus one can be given for any
micro-state, while such observables can be realized as classical observables
with sharp values on a substate level. For a continuous family of micro-states
parameterized by a sphere all the quantum mechanical laws for a two-state
system follow under the assumption that the purity of the ensemble is conserved
by the time evolution. The non-commutative correlation functions of quantum
mechanics correspond to the use of conditional correlation functions in
classical statistics. We further discuss the classical statistical realization
of entanglement within a system corresponding to four-state quantum mechanics.
We conclude that quantum mechanics can be derived from a classical statistical
setting with infinitely many micro-states.
