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S13f. Objective probabilities
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Consider a physical die (for simplicity assumed perfectly symmetric)
with six elementary events 1,...,6.
If the die is not thrown, all events are equivalent, and the
probabilities are 1/6 for each event. These probabilities are
associated to the die (_not_ to a throw), and can be determined
uniquely from the knowledge of the geometry and composition of
the die. All of probability theory happens at this level,
since the 'happening' of an event is not formally defined.
If the die is thrown, a given event (say 3) either happens or
does not happen. If the event happens (does not happen), the
statement 'This throw is a 3' is true (false), hence has a
probability of 100% (0%), although before the throw, these
probabilities are not yet known. These probabilities are
associated to each particular throw (_not_ to the die).
Thus a die functions as a potential stationary source of throws,
and hence _defines_ an ensemble of (conceivable) throws.
An actual throw, though a realization of this ensemble,
is determined by the outcome, and cannot be assigned a
probability different from 0 or 1.
[See, e.g., the wikipedia entry
http://en.wikipedia.org/wiki/Probability_theory
''Omega is a non-empty set, sometimes called the "sample space",
each of whose members is thought of as a potential outcome of a
random experiment.''
'is thought of' signifies the interpretational level.
Probabilities are only about 'potential outcomes' (what I call
conceivable), not abut actual ones.]
A stationary source has objective probability distributions
for random vectors computable from observations made on it.
These are given in terms of an objective expectation mapping
and an associated density. In principle, this density can be
measured arbitrarily well, and if the form and composition of
the source is known, can be objectively predicted from
physical theories.
Thus objective probability distributions exist always when the
generating ensemble is completely known, and more generally
whenever it is objectively determined.
Similarly, in quantum theory, a laser is a potential stationary source
of photons, the oven in a Stern-Gerlach experiment is a stationary
source of electrons, etc. The sources are in well-defined,
objective quantum mechanical states, defining ensembles with
objectively predictable properties.