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Concept
Empirical probability
Summary
In probability theory and statistics, the empirical probability, relative frequency, or experimental probability of an event is the ratio of the number of outcomes in which a specified event occurs to the total number of trials, i.e., by means not of a theoretical sample space but of an actual experiment. More generally, empirical probability estimates probabilities from experience and observation.
Given an event A in a sample space, the relative frequency of A is the ratio \tfrac m n, m being the number of outcomes in which the event A occurs, and n being the total number of outcomes of the experiment.
In statistical terms, the empirical probability is an estimator or estimate of a probability. In simple cases, where the result of a trial only determines whether or not the specified event has occurred, modelling using a binomial distribution might be appropriate and then the empirical estimate is the maximum likelihood estimate. It is the Bayesian estimate for the same case if certain assumptions are made for the prior distribution of the probability. If a trial yields more information, the empirical probability can be improved on by adopting further assumptions in the form of a statistical model: if such a model is fitted, it can be used to derive an estimate of the probability of the specified event
An advantage of estimating probabilities using empirical probabilities is that this procedure is relatively free of assumptions.
For example, consider estimating the probability among a population of men that they satisfy two conditions:
that they are over 6 feet in height.
that they prefer strawberry jam to raspberry jam.
A direct estimate could be found by counting the number of men who satisfy both conditions to give the empirical probability of the combined condition. An alternative estimate could be found by multiplying the proportion of men who are over 6 feet in height with the proportion of men who prefer strawberry jam to raspberry jam, but this estimate relies on the assumption that the two conditions are statistically independent.
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