The word probability derives from the Latin probare (to prove, or to test). Informally, probable is one of several words applied to uncertain events or knowledge, being more or less interchangeable with likely, risky, hazardous, uncertain, and doubtful, depending on the context. Chance, odds, and bet are other words expressing similar notions. As with the theory of mechanics which assigns precise definitions to such everyday terms as work and force< ...

Including:

• Probability - Historical remarks
• Probability - Concepts
• Probability - Formalization of probability
• Probability - Representation and interpretation of probability values
• Probability - Distributions
• Probability - Probability in mathematics
• Probability - Remarks on probability calculations
• Probability - Applications of probability theory to everyday life
• Probability - Quotations

Read more here: » Probability: Encyclopedia - Probability

See also:

Bayesian probability, Bayesian probability - Controversy, Bayesian probability - History of Bayesian probability, Bayesian probability - Varieties of Bayesian probability, Bayesian probability - Bayesian and frequentist probability, Bayesian probability - Applications of Bayesian probability, Bayesian probability - Probabilities of probabilities

Read more here: » Bayesian probability: Encyclopedia II - Bayesian probability - Probabilities of probabilities

Like other theories, the theory of probability is a representation of probabilistic concepts in formal terms -- that is, in terms that can be considered separately from their meaning. These formal terms are manipulated by the rules of mathematics and logic, and any results are then interpreted or translated back into the problem domain. There have been at least two successful attempts to formalize probability, namely the Kolmogorov formulation and the Cox formulation. In Kolmogorov's formulation, sets are interpreted as events and pro ...

See also:

Probability, Probability - Historical remarks, Probability - Concepts, Probability - Formalization of probability, Probability - Representation and interpretation of probability values, Probability - Distributions, Probability - Probability in mathematics, Probability - Remarks on probability calculations, Probability - Applications of probability theory to everyday life, Probability - Quotations

Read more here: » Probability: Encyclopedia II - Probability - Formalization of probability

If a female gives birth to a haemophiliac child, she is possibly a carrier for the disease. Until modern direct DNA testing, however, it was impossible to determine if a female with only healthy children was a carrier or not. Generally, the more healthy sons she bore, the higher the probability that she was not a carrier, specifically where x is the number of unaffected sons. (More complicated formulae could be used if healthy grandchildren and other rela ...

See also:

Haemophilia, Haemophilia - Forms, Haemophilia - Genetics, Haemophilia - Probability, Haemophilia - Table, Haemophilia - Treatment, Haemophilia - History

Read more here: » Haemophilia: Encyclopedia II - Haemophilia - Probability

If a female gives birth to a haemophiliac child, she is possibly a carrier for the disease. Until modern direct DNA testing, however, it was impossible to determine if a female with only healthy children was a carrier or not. Generally, the more healthy sons she bore, the higher the probability that she was not a carrier, specifically where x is the number of unaffected sons. (More complicated formulae could be used if healthy grandchildren and other relatives were to be taken into consideration.) It is estimated that about 0.006% percent of the United ...

See also:

Haemophilia, Haemophilia - Forms, Haemophilia - Genetics, Haemophilia - Probability, Haemophilia - Table, Haemophilia - Treatment, Haemophilia - History

Read more here: » Haemophilia: Encyclopedia II - Haemophilia - Probability

Statistics makes extensive use of the concept of probability. The probability of an event is often defined as a number between one and zero. In reality however there is virtually nothing that has a probability of 1 or 0. You could say that the sun will certainly rise in the morning, but what if an extremely unlikely event destroys the sun? What if there is a nuclear war and the sky is covered in ash and smoke? We often round the probability of such things up or down because they are so likely or unlikely to occur, that it's easier to recog ...

See also:

Statistics, Statistics - Origin, Statistics - Statistical methods, Statistics - Experimental and observational studies, Statistics - Levels of measurement, Statistics - Statistical techniques, Statistics - Probability, Statistics - Important contributors to statistics, Statistics - Specialized disciplines, Statistics - Software, Statistics - Additional references

Read more here: » Statistics: Encyclopedia II - Statistics - Probability

The scientific study of probability is a modern development. Gambling shows that there has been an interest in quantifying the ideas of probability for millennia, but exact mathematical descriptions of use in those problems only arose much later. The doctrine of probabilities dates to the correspondence of Pierre de Fermat and Blaise Pascal (1654). Christiaan Huygens (1657) gave the earliest known scientific treatment of the subject. Jakob Bernoulli's Ars Conjectandi (posthumous, 1713) and Abraham de Moivre's Doctrine of Chances (171 ...

See also:

Probability, Probability - Historical remarks, Probability - Concepts, Probability - Formalization of probability, Probability - Representation and interpretation of probability values, Probability - Distributions, Probability - Probability in mathematics, Probability - Remarks on probability calculations, Probability - Applications of probability theory to everyday life, Probability - Quotations

Read more here: » Probability: Encyclopedia II - Probability - Historical remarks

Around 1960, Ray Solomonoff invented the concept of algorithmic probability. Take a universal computer and randomly generate an input program. The program will compute some possibly infinite output. The algorithmic probability of any given finite output prefix q is the sum of the probabilities of the programs that compute something starting with q. Certain long objects with short programs have high probability. Algorithmic probability is the main ingredient of Ray Solomonoff's theory of inductive inference, the theory of ...

Read more here: » Algorithmic probability: Encyclopedia - Algorithmic probability

The appeal to probability is a logical fallacy, often used in conjunction with other fallacies. It assumes that because something could happen, it is inevitable that it will happen. This is flawed logic, regardless of the likelihood of the event in question. The fallacy is often used to exploit paranoia. This has the argument form: Possibly P. Therefore, P is true. Equivalently, using modal logi ...

Read more here: » Appeal to probability: Encyclopedia - Appeal to probability

Bayesianism is the philosophical tenet that the mathematical theory of probability applies to the degree of plausibility of a statement. This also applies to the degree of believability contained within the rational agents of a truth statement. Additionally, when a statement is used with Bayes' theorem, it then becomes a Bayesian inference. This is in contrast to frequentism, which rejects degree-of-belief interpretations of mathematical probability, and assigns probabilities only to random events according to their relative fr ...

Including:

• Bayesian probability - History of Bayesian probability
• Bayesian probability - Varieties of Bayesian probability
• Bayesian probability - Bayesian and frequentist probability
• Bayesian probability - Applications of Bayesian probability
• Bayesian probability - Bayesian data analysis

Read more here: » Bayesian probability: Encyclopedia - Bayesian probability

This article defines some terms which characterize probability distributions of two or more variables. Conditional probability is the probability of some event A, given that some other event, B, has already occurred. Conditional probability is written P(A|B), and is read "the probability of A, given B". Joint probability is the proba ...

Including:

• Conditional probability - Definition
• Conditional probability - Statistical independence
• Conditional probability - Mutual exclusivity
• Conditional probability - Other considerations

Read more here: » Conditional probability: Encyclopedia - Conditional probability

The problems and paradoxes of the classical interpretation of probability motivated the development of the relative frequency concept of probability. Most of the mathematics commonly used to make statistical estimates or tests are developed by statisticians who use this concept exclusively. They are usually called frequentists, and their position is called frequentism. A statistician who uses traditional methods of inference is therefore referred to as a frequentist statistician. Frequentism is, by far, the most commonly held view among ...

Including:

• Frequency probability - Bibliography

Read more here: » Frequency probability: Encyclopedia - Frequency probability

In probability theory, the characteristic function of any probability distribution on the real line is given by the following formula, where X is any random variable with the distribution in question: where t is a real number and E denotes the expected value. If FX is the cumulative distribution function, then the characteristic function is given by the Riemann-Stieltjes integral In cases in which there is a probabili ...

Including:

• Characteristic function probability theory - The inversion theorem
• Characteristic function probability theory - The continuity theorem
• Characteristic function probability theory - Uses of characteristic functions
• Characteristic function probability theory - Related concepts

Read more here: » Characteristic function probability theory: Encyclopedia - Characteristic function probability theory

The Wigner quasi-probability distribution was introduced by Eugene Wigner in 1932 to study quantum corrections to classical statistical mechanics. The goal was to replace the wavefunction that appears in Schrodinger's equation with a probability distribution in phase space. It was independently derived by Hermann Weyl in 1931 as the symbol of the density matrix in representation theory in mathematics. It was once again derived by J. Ville in 1948 as a quadratic (in signal) representation of the local time-frequency energy of a signal. ...

Including:

• Wigner quasi-probability distribution - Mathematical properties
• Wigner quasi-probability distribution - Uses of the Wigner function outside quantum mechanics
• Wigner quasi-probability distribution - Measurements of the Wigner function
• Wigner quasi-probability distribution - Other related quasi-probability distributions
• Wigner quasi-probability distribution - Historical note

Read more here: » Wigner quasi-probability distribution: Encyclopedia - Wigner quasi-probability distribution

From the Kolmogorov axioms one can deduce other useful rules for calculating probabilities: This is called The Addition Law of Probability. That is, the probability that A or B will happen is the sum of the probabilities that A will happen and that B will happen, minus the probability that A and B will happen. This can be extended to the inclusion-exclusion principle. That is, the probability that any event will not happen is 1 minus the probability that it will. Using conditional probability as defined ...

See also:

Probability axioms, Probability axioms - Kolmogorov axioms, Probability axioms - First axiom, Probability axioms - Second axiom, Probability axioms - Third axiom, Probability axioms - Lemmas in probability

Read more here: » Probability axioms: Encyclopedia II - Probability axioms - Lemmas in probability

From the Kolmogorov axioms one can deduce other useful rules for calculating probabilities: This is called the additition law of probability. That is, the probability that A or B will happen is the sum of the probabilities that A will happen and that B will happen, minus the probability that A and B will happen. This can be extended to the inclusion-exclusion principle. That is, the probability that any event wil ...

See also:

Probability axioms, Probability axioms - Kolmogorov axioms, Probability axioms - First axiom, Probability axioms - Second axiom, Probability axioms - Third axiom, Probability axioms - Lemmas in probability

Read more here: » Probability axioms: Encyclopedia II - Probability axioms - Lemmas in probability

The terms subjective probability, personal probability, epistemic probability and logical probability describe some of the schools of thought which are customarily called "Bayesian". These overlap but there are differences of emphasis. Some of the people mentioned here would not call themselves Bayesians. Bayesian probability is supposed to measure the degree of belief an individual has in an uncertain proposition, and is in that respect subjective. Some people who call themselves Bayesians do not accept th ...

See also:

Bayesian probability, Bayesian probability - Controversy, Bayesian probability - History of Bayesian probability, Bayesian probability - Varieties of Bayesian probability, Bayesian probability - Bayesian and frequentist probability, Bayesian probability - Applications of Bayesian probability, Bayesian probability - Probabilities of probabilities

Read more here: » Bayesian probability: Encyclopedia II - Bayesian probability - Varieties of Bayesian probability

Today, there are a variety of applications of Bayesian probability that have gained wide acceptance. Some schools of thought emphasise Cox's theorem and Jaynes' principle of maximum entropy as cornerstones of the theory, others (e.g., Ramsey, di Finetti) approach it from the point of view of a Dutch book argument, still others may claim that Bayesian methods are more general and give better results in practice than frequency probability. Se ...

See also:

Bayesian probability, Bayesian probability - Controversy, Bayesian probability - History of Bayesian probability, Bayesian probability - Varieties of Bayesian probability, Bayesian probability - Bayesian and frequentist probability, Bayesian probability - Applications of Bayesian probability, Bayesian probability - Probabilities of probabilities

Read more here: » Bayesian probability: Encyclopedia II - Bayesian probability - Applications of Bayesian probability

The Bayesian approach is in contrast to the concept of frequency probability where probability is held to be derived from observed or predicted frequency distributions or proportions of populations, with the usefulness of probability narrowly limited to such scenarios. The difference has many implications for the methods by which statistics is practiced when following one model or the other, and also for the way in which conclusions are expressed. For example, Laplace estimated the mass of Saturn using Bayesian methods. However ...

See also:

Bayesian probability, Bayesian probability - Controversy, Bayesian probability - History of Bayesian probability, Bayesian probability - Varieties of Bayesian probability, Bayesian probability - Bayesian and frequentist probability, Bayesian probability - Applications of Bayesian probability, Bayesian probability - Probabilities of probabilities

Read more here: » Bayesian probability: Encyclopedia II - Bayesian probability - Bayesian and frequentist probability

"Bayesian" probability or "Bayesian" theory is named after Thomas Bayes (1701? — 1761), who proved a special case of what is called Bayes' theorem. The term Bayesian, however, came into use only around 1950, and in fact it is not clear that Bayes would have endorsed the very broad interpretation of probability now called "Bayesian." Laplace independently proved a more general version of Bayes' theorem and put it to good use in solving problems in celestial mechanics, medical statistics and, by some accounts, even jurisprudence. Lapl ...

See also:

Bayesian probability, Bayesian probability - Controversy, Bayesian probability - History of Bayesian probability, Bayesian probability - Varieties of Bayesian probability, Bayesian probability - Bayesian and frequentist probability, Bayesian probability - Applications of Bayesian probability, Bayesian probability - Probabilities of probabilities

Read more here: » Bayesian probability: Encyclopedia II - Bayesian probability - History of Bayesian probability