Relative frequencies from beliefs

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 $n$ independent trials, for each of which we believe that the `success' will occur with probability $p$, the expected number of successes is $np$, with a standard uncertainty $\sqrt{np(1-p)}$. We expect then a relative frequency $p$ [that is $(np)/n$] with an uncertainty $\sqrt{p(1-p)/n}$ [that is $\sqrt{np(1-p)}/n$]. When $n$ is very large, the uncertainty goes to zero and we become `practically sure' to observe a relative frequency very close to $p$. 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