Classical Interpretation

The classical interpretation goes back to Laplace. According to his definition the probability of an outcome is the ratio of favorable cases to the number of equally possible cases. Two outcomes are meant ``equally possible'' if we have no reason to prefer one to the other (principle of indifference). Consider the following often quoted example: A symmetric die has six faces numbered 1-6. When it is tossed in the standard way there are six possible outcomes. The probability of getting an <even number> is $\frac{3}{6}$, for three of the possible outcomes (2, 4, 6) are favorable.

Since we are talking about a realistic interpretation, we must translate the probability-theoretic notions into an observational language. But how can we perform such a translation into empirical terms? In no way, it seems. For either we obtain a concept of probability which is brutally indifferent to the facts of the world, or we must tacitly refer to another, for instance the frequency interpretation of probability. In the case of the above example of the symmetric die, the essential fact is that in the moment of tossing the die the history of the universe is (objectively or epistemically, all the same) branching into six branches.

More precisely, the possible histories can be sorted into six classes corresponding to the outcomes 1-6. There are three branches that correspond to the event <even number>. Therefore, $p\left(<\textrm{event number}>\right)=\frac{3}{6}$. All other facts of the world are negligible. For instance, if the die were biased, the probability of getting an even number would be the same. 'But a biased die is not symmetric anymore, - we could argue - so one cannot apply the principle of indifference in the same way!' This argument is, however, problematic, because it appeals to two different conceptions of probability. Of course, the biased die is not symmetric in respect of some properties. The mass-density, for example, is not symmetric. But even a standard die is not completely symmetric in all properties. Otherwise we were not able to differentiate the six outcomes. For instance, different numbers are written on the different faces of the die. So, it seems, we must conclude, that there are relevant and irrelevant asymmetries: only those asymmetries are relevant, which can influence the probabilities of the six outcomes. 'But what kind of ``probability'' do we mean here?', we should ask ourselves. And the only possible answer could be something like this: It is an observable fact that the biased die produces one outcome more oftenthan the other. That is, we should refer to the frequency interpretation of probability.