Empirical probability

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,^[1] 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.^[2]

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.^[3]

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

^ Mood, A. M.; Graybill, F. A.; Boes, D. C. (1974). "Section 2.3". Introduction to the Theory of Statistics (3rd ed.). McGraw-Hill. ISBN 0070428646.
^ "Empirical probabilities at tpub.com". Archived from the original on 2007-05-10. Retrieved 2007-03-31.
^ Gujarati, Damodar N. (2003). "Appendix A". Basic Econometrics (4th ed.). McGraw-Hill. ISBN 978-0-07-233542-2.

[Mood-1] Mood, A. M.; Graybill, F. A.; Boes, D. C. (1974). "Section 2.3". Introduction to the Theory of Statistics (3rd ed.). McGraw-Hill. ISBN 0070428646.

[2] "Empirical probabilities at tpub.com". Archived from the original on 2007-05-10. Retrieved 2007-03-31.

[3] Gujarati, Damodar N. (2003). "Appendix A". Basic Econometrics (4th ed.). McGraw-Hill. ISBN 978-0-07-233542-2.

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