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S13d. Are probabilities limits of relative frequencies?
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Sometimes, probabilities are regarded as limits of relative
frequencies as the number of trials becomes arbitrarily large.
But the weak law of large numbers only guarantees that most trial
histories will give a sequence of relative frequencies that converge
to the probability. It might just fail for the one actually tried...
Moreover, in practice we only have partial knowledge of such an infinite
sequence of trials (which cannot be performed). This knowledge about
the sample give no knowledge at all about the limiting ensemble.
Just as the knowledge of the first n items of a sequence give, in
theory, no knowledge at all about the limit of the sequence.
That we often estimate the limit using a small part of the sequence
is asnother matter, and is like estimating probabilities from samples.
But the estimate may be completely wrong.
Thus interpreting probability as relative frequency is a philosophically
difficult interpretation step. For a thorough discussion, see the very
informative books by
T.L. Fine,
Theory of probability; an examination of foundations.
Acad. Press, New York 1973.
and
L. Sklar,
Physics and Chance,
Cambridge Univ. Press, Cambridge 1993.