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 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 ...

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 ...

it is applied in the discrete uniform distribution 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 ergodic ...

for rational numbers. If there are

n

events under consideration, the probability of each occurring is

\frac.

In philosophy it corresponds to a concept that allows one to assign equal probabilities 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'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

N

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

\frac.

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, 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

modeling A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...

. 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 ...

, one needs to establish

prior probabilities In Bayesian probability, 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 e ...

for the various hypotheses before applying Bayes' theorem. 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.

References

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External links

Quotes on equiprobability
in classical probability Probability interpretations Philosophy of statistics

See also

References

External links