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##

What is probability?

The standard answers to this question are
- ``the ratio of the number of favourable cases to the
number of all cases'';
- ``the ratio of the number of times the event occurs in a test series
to the total number of trials in the series''.

It is very easy to show that neither of these
statements can define the concept of probability:
- Definition (1) lacks
the clause ``if all the cases are
__equally probable__''. This has
been done here intentionally, because people often forget it.
The fact that the definition of probability makes use of the term
``probability'' is clearly embarrassing. Often in textbooks the
clause is replaced by ``if all the cases are
equally possible'', ignoring that in this
context ``possible''
is just a synonym of ``probable''. There is no way out.
This statement does not
define probability but
gives, at most, a useful rule for evaluating it -
assuming we
know what probability is, i.e. of what we are talking about.
The fact that this definition is labelled
``classical'' or ``Laplace'' simply shows that some
authors are not
aware of what the ``classicals'' (Bayes, Gauss, Laplace, Bernoulli, etc.)
thought about this matter. We shall call this ``definition''
*combinatorial*.
- Definition (2) is also incomplete, since it lacks the condition
that the number of trials must be very large (``it goes to infinity'').
But this is a minor point. The crucial point is that the
statement merely defines the relative
*frequency* with
which an event
(a ``phenomenon'')
occurred in the past. To use frequency as a measurement of
probability we have to assume that the phenomenon
occurred in the past, and will occur in the future,
__with the same probability__. But who can tell if this hypothesis
is correct? Nobody: __we__
have to guess in every single case. Note that, while in the
first ``definition'' the assumption of equal probability
was explicitly stated, the analogous clause is often
missing from the second one. We shall call this ``definition''
*frequentistic*.

We have to conclude that if we want to make use of these
statements
to assign a numerical value to probability, in those cases
in which __we judge__ that the clauses are satisfied, we need
a better definition of probability.

** Next:** Subjective definition of probability
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Giulio D'Agostini
2003-05-15