  In the paper the basic concepts of extended probability theory are
introduced. The basic idea: the concept of an event as a subset of \Omega is
replaced with the concept of an event as a partition. The partition is any set
of disjoint non-empty subsets of \Omega (i.e. partition=subset+its
decomposition). Interpretation: elements inside certain part are
indistinguishable, while elements from different parts are distinguishable.
There are incompatible events, e.g {{e1},{e2}} and {{e1,e2}}. This is logical
incompatibility analogical to the impossibility to have and simultaneously not
to have the which-way information in the given experiment. The context is the
maximal set of mutually compatible events. Each experiment has associated its
context. In each context the extended probability is reduced to classical
probability. Then the quadratic representation of events, partitions and
probability measures is developed. At the end the central concept of quadratic
probability spaces (which extend Kolmogorov probability spaces) is defined and
studied. In the next paper it will be shown that quantum mechanics can be
represented as the theory of Markov processes in the extended probability
theory (Einstein's vision of QM).
