Define: Empirical Probability

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Definition: Empirical Probability


Quick Summary of Empirical Probability


In probability theory, the empirical probability is an estimated probability based upon previous evidence or experimental results. As such, the empirical probability is sometimes referred to as experimental probability, and we can distinguish it from probabilities calculated from a clearly-defined sample space. The empirical probability, relative frequency, or experimental probability of an event is the ratio of the number of outcomes in




What is the dictionary definition of Empirical Probability?

Dictionary Definition


The empirical probability of an event is the ratio of the number of outcomes in which a specified event occurs to the total number of trials, not in a theoretical sample space but in an actual experiment.

Empirical Probability is sometimes also referred to as: relative frequency or experimental probability


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Full Definition of Empirical Probability


In probability theory, the empirical probability is an estimated probability based upon previous evidence or experimental results. As such, the empirical probability is sometimes referred to as experimental probability, and we can distinguish it from probabilities calculated from a clearly-defined sample space.

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, not in a theoretical sample space but in an actual experiment. In a more general sense, the empirical probability estimates probabilities from observation and experience.

Given an event A in a sample space, the relative frequency of A is the ratio 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 estimate or estimator 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.

Understanding Empirical Probability

In order for a theory to be proved or disproved, empirical evidence must be collected. An empirical study will be performed using actual market data. For example, many empirical studies have been conducted on the capital asset pricing model (CAPM), and the results are slightly mixed.

In some analyses, the model does hold in real-world situations, but most studies have disproved the model for projecting returns. Although the model is not completely valid, that is not to say that there is no utility associated with using the CAPM. For instance, the CAPM is often used to estimate a company’s weighted average cost of capital.

Mixed nomenclature

The phrase a-posteriori probability is also used as an alternative to empirical probability or relative frequency. The use of the phrase “a-posteriori” is reminiscent of terms in Bayesian statistics, but is not directly related to Bayesian inference, where a-posteriori probability is occasionally used to refer to posterior probability, which is different even though it has a confusingly similar name.

The term a-posteriori probability, in its meaning as equivalent to empirical probability, may be used in conjunction with a priori probability which represents an estimate of a probability not based on any observations but based on deductive reasoning


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Empirical Probability FAQ's


What is Empirical Probability?

Empirical probability uses the number of occurrences of an outcome within a sample set as a basis for determining the probability of that outcome. The number of times “event X” happens out of 100 trials will be the probability of event X happening. An empirical probability is closely related to the relative frequency of an event.

What is the advantages of using Empirical Probability?

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 5 feet in height.
that they prefer dark chocolate to white chocolate.

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 5 feet in height with the proportion of men who prefer dark chocolate to white chocolate, but this estimate relies on the assumption that the two conditions are statistically independent.

What are the disadvantages of using Empirical Probability?

A disadvantage in using empirical probabilities arises in estimating probabilities which are either very close to zero or very close to one. In these cases, very large sample sizes would be needed in order to estimate such probabilities to a good standard of relative accuracy. Here statistical models can help, depending on the context, and in general, one can hope that such models would provide improvements in accuracy compared to empirical probabilities, provided that the assumptions involved actually do hold.

For example, consider estimating the probability that the lowest of the daily-maximum temperatures at a site in February in any one year is less than zero degrees Celsius. A record of such temperatures in past years could be used to estimate this probability. A model-based alternative would be to select a family of probability distributions and fit it to the dataset containing past years′ values. The fitted distribution would provide an alternative estimate of the desired probability. This alternative method can provide an estimate of the probability even if all values in the record are greater than zero.


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This glossary post was last updated: 9th August 2019.