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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
Up: Probability
Previous: Probability
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Giulio D'Agostini
2003-05-15