What is probability?

The standard answers to this question are

1. ``the ratio of the number of favourable cases to the number of all cases'';
2. ``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.