The question of how relative frequencies of occurrence
follow from beliefs is much easier. It is
a simple consequence of probability theory and can
be easily understood by anyone familiar with the binomial
distribution, taught in any elementary course on probability.
If we think at independent trials, for each of which
we believe that the `success' will occur with probability ,
the *expected number* of successes is , with
a *standard uncertainty*
.
We expect then a relative frequency [that is ]
with an uncertainty
[that is
]. When is very large,
the uncertainty goes to zero and
we become `practically sure' to observe a relative frequency
very close to . This asymptotic feature goes under the name
of *Bernoulli theorem*. It is important to remark that
this reasoning
can be *purely hypothetical* and has nothing to do
with the so called frequentistic definition of probability.
To conclude this section, probabilities can be evaluated from
(past) frequencies and (future, or hypothetical)
frequencies can be evaluated from probabilities, but
*probability is not frequency*.^{42}

Giulio D'Agostini
2010-09-30