We need to make a distinction between *mathematical* and *realistic*
interpretations of probability. Mathematical interpretation means
that the formal mathematical structure PROBABILITY THEORY
is represented in ANOTHER FORMAL MATHEMATICAL STRUCTURE.
Discussing the Bertrand-paradox, for example, we consider a representation
of the probability-theoretic notions in geometrical terms. (The alleged
``paradox'' consists in the simple fact that we have a kind of
freedom in constructing such a representation.)

Problem:Given a circle. Find the probability that a chord chosen at random be longer than the side of an inscribed equilateral triangle.

Solution 1

Let us fix the point A and consider only the chords that emanate from this point. Then it becomes clear that 1/3 of the outcomes will result in a chord longer than the side of an equilateral triangle.

Solution 2

It is also a mathematical representation when probabilities are represented by the limiting values of convergent relative-frequency-like infinite sequences, or when the (``subjective'') probabilities are represented in game-theoretic terms.A chord is fully determined by its midpoint. Chords whose length exceeds the side of the triangle have their midpoints inside a smaller circle with radius . Hence, .

The *mathematical interpretations do not raise difficulties at
all*. Our concern is, however, not a mathematical but a realistic
interpretation. A realistic interpretation is nothing but the way
in which we apply probability theory to the real world. The similarity
between the mathematical and realistic interpretations is that in
both cases we construct a representation of PROBABILITY THEORY
in ANOTHER LANGUAGE. But, in case of a realistic interpretation
this other language *must be, in final analysis, translatable
into empirical terms*.

In this section I would like to briefly review the standard interpretations,^{3} and to illustrate that none of them is tenable, because none of them
provides a sound definition of what probability is.

- Classical Interpretation
- Relative Frequency Interpretation
- Propensity Interpretation
- Subjective Interpretation

2003-10-23