Equiprobability is a property for a collection of events that each have the same
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
of occurring.
In
statistics and
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
it is applied in the
discrete uniform distribution
In probability theory and statistics, the discrete uniform distribution is a symmetric probability distribution wherein a finite number of values are equally likely to be observed; every one of ''n'' values has equal probability 1/''n''. Ano ...
and the
equidistribution theorem
In mathematics, the equidistribution theorem is the statement that the sequence
:''a'', 2''a'', 3''a'', ... mod 1
is uniformly distributed on the circle \mathbb/\mathbb, when ''a'' is an irrational number. It is a special case of the ergodi ...
for rational numbers. If there are
events under consideration, the probability of each occurring is
In
philosophy it corresponds to a concept that allows one to assign equal
probabilities
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...
to outcomes when they are judged to be
equipossible
Equipossibility is a philosophical concept in possibility theory that is a precursor to the notion of equiprobability in probability theory. It is used to distinguish what ''can'' occur in a probability experiment. For example, it is the differenc ...
or to be "equally likely" in some sense. The best-known formulation of the rule is
Laplace
Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summariz ...
's
principle of indifference
The principle of indifference (also called principle of insufficient reason) is a rule for assigning epistemic probabilities. The principle of indifference states that in the absence of any relevant evidence, agents should distribute their cre ...
(or ''principle of insufficient reason''), which states that, when "we have no other
information
Information is an abstract concept that refers to that which has the power to inform. At the most fundamental level information pertains to the interpretation of that which may be sensed. Any natural process that is not completely random, ...
than" that exactly
mutually exclusive
In logic and probability theory, two events (or propositions) are mutually exclusive or disjoint if they cannot both occur at the same time. A clear example is the set of outcomes of a single coin toss, which can result in either heads or tails ...
events can occur, we are justified in assigning each the probability
This
subjective assignment of probabilities is especially justified for situations such as
rolling dice and
lotteries
A lottery is a form of gambling that involves the drawing of numbers at random for a prize. Some governments outlaw lotteries, while others endorse it to the extent of organizing a national or state lottery. It is common to find some degree of ...
since these experiments carry a
symmetry structure, and one's state of knowledge must clearly be
invariant under this symmetry.
A similar argument could lead to the seemingly absurd conclusion that the sun is as likely to rise as to not rise tomorrow morning. However, the conclusion that the sun is equally likely to rise as it is to not rise is only absurd when additional information is known, such as the laws of gravity and the sun's history. Similar applications of the concept are effectively instances of
circular reasoning
Circular may refer to:
* The shape of a circle
* ''Circular'' (album), a 2006 album by Spanish singer Vega
* Circular letter (disambiguation)
** Flyer (pamphlet), a form of advertisement
* Circular reasoning, a type of logical fallacy
* Circul ...
, with "equally likely" events being assigned equal probabilities, which means in turn that they are equally likely. Despite this, the notion remains useful in probabilistic and
statistical
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industr ...
modeling.
In
Bayesian probability
Bayesian probability is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification o ...
, one needs to establish
prior probabilities for the various hypotheses before applying
Bayes' theorem
In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For exa ...
. One procedure is to assume that these prior probabilities have some symmetry which is typical of the experiment, and then assign a prior which is proportional to the
Haar measure for the symmetry group: this generalization of equiprobability is known as the
principle of transformation groups The principle of transformation groups is a rule for assigning ''epistemic'' probabilities in a statistical inference problem. It was first suggested by Edwin T. Jaynes and can be seen as a generalisation of the principle of indifference.
This can ...
and leads to misuse of equiprobability as a model for incertitude.
See also
*
Principle of indifference
The principle of indifference (also called principle of insufficient reason) is a rule for assigning epistemic probabilities. The principle of indifference states that in the absence of any relevant evidence, agents should distribute their cre ...
*
Laplacian smoothing
Laplacian smoothing is an algorithm to smooth a polygonal mesh. For each vertex in a mesh, a new position is chosen based on local information (such as the position of neighbours) and the vertex is moved there. In the case that a mesh is topolo ...
*
Uninformative prior
In Bayesian statistical inference, a prior probability distribution, often simply called the prior, of an uncertain quantity is the probability distribution that would express one's beliefs about this quantity before some evidence is taken int ...
*
A priori probability
An ''a priori'' probability is a probability that is derived purely by deductive reasoning. One way of deriving ''a priori'' probabilities is the principle of indifference, which has the character of saying that, if there are ''N'' mutually excl ...
*
Aequiprobabilism
Aequiprobabilism, also spelled æquiprobabilism or equiprobabilism, is one of several doctrines in moral theology opposed to probabilism.
Teaching
#If the opinions for and against the current existence of a law have equal or nearly equal probabil ...
*Uniform probability distributions:
**
Continuous
**
Discrete
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
*Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a ...
*
Information gain
Information is an abstract concept that refers to that which has the power to inform. At the most fundamental level information pertains to the interpretation of that which may be sensed. Any natural process that is not completely random, ...
References
{{reflist
External links
Quotes on equiprobabilityin classical probability
Probability interpretations
Philosophy of statistics