Frequentist Probability - Definition

Definition

In the frequentist interpretation, probabilities are discussed only when dealing with well-defined random experiments. The set of all possible outcomes of a random experiment is called the sample space of the experiment. An event is defined as a particular subset of the sample space to be considered. For any given event, only one of two possibilities may hold: it occurs or it does not. The relative frequency of occurrence of an event, observed in a number of repetitions of the experiment, is a measure of the probability of that event. This is the core conception of probability in the frequentist interpretation.

Thus, if is the total number of trials and is the number of trials where the event occurred, the probability of the event occurring will be approximated by the relative frequency as follows:

Clearly, as the number of trials is increased, one might expect the relative frequency to become a better approximation of a "true frequency".

A controversial claim of the frequentist approach is that in the "long run," as the number of trials approaches infinity, the relative frequency will converge exactly to the true probability:

Such a limit is possible only in mathematical settings, where accurate repetitions unto infinity may be made (e.g. such as counting the relative fraction of even numbers less than n_t: one may easily compute the limit .) This conflicts with the standard claim that the frequency interpretation is somehow more "objective" than other theories of probability.