The frequency interpretation is based on the following idea: Probability of an outcome is not a concept which could be assigned to an individual experiment, but rather it is assigned to a long sequence of repeated experiments. Denote the Boolean algebra of the outcome events, and let be the probability to be interpreted in empirical terms. Performing an experiment, each of the possible outcome events does or does not occur, and this fact can be described by a suitable ``outcome function'' (classical two-valued truth function, if you want) satisfying the following conditions:
Consider now the sequential repetitions of an experiment.
One can easily prove that if the sequence
Here we encounter the first difficulty of the Frequency Interpretation: the limit of an infinite sequence is independent of the first elements, where can be an arbitrarily large number. In other words, the observed relative frequency in any finite sample is irrelevant to the probability. How, then, we are supposed to find out what these probabilities are? Or if we have a hypotheses about the value of a probability, how can we empirically confirm this hypotheses, if there is no logical relationship between what we observe in a finite sample and the value we would like to confirm?
The second difficulty is that
This conclusion is, however, counter-intuitive, because we ``know'' that in each run of the experiment probabilities and do exist, for instance for a while, then it changes for , then changes for again, and so on.
The above example also throws light on the third problem of the Frequency Interpretation. Namely, that it does not account for the probability of the outcome of an individual experiment, which probability, on the other hand, is a meaningful concept in our intuition.