## English

### Noun

probabilities (plural)- Plural of probability

Probability is the likelihood or chance that
something is the case or will happen. Probability
theory is used extensively in areas such as statistics, mathematics, science and philosophy to draw
conclusions about the likelihood of potential events and the
underlying mechanics of complex systems.

The word probability does not have a consistent
direct definition. Actually, there are two broad categories of
probability interpretations: Frequentists talk about
probabilities only when dealing with well defined random experiments. The
relative frequency of occurrence of an experiment's outcome, when
repeating the experiment, is a measure of the probability of that
random event. Bayesians,
however, assign probabilities to any statement whatsoever, even
when no random process is involved, as a way to represent its
subjective plausibility.

According to Richard Jeffrey, "Before the middle
of the seventeenth century, the term 'probable' (Latin probabilis)
meant approvable, and was applied in that sense, univocally, to
opinion and to action. A probable action or opinion was one such as
sensible people would undertake or hold, in the
circumstances."

Aside from some elementary considerations made by
Girolamo
Cardano in the 16th century, 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 (1718) treated the subject as a branch of
mathematics. See Ian Hacking's
The Emergence of Probability for a history of the early development
of the very concept of mathematical probability.

The theory of errors may be traced back to
Roger
Cotes's Opera Miscellanea (posthumous, 1722), but a memoir
prepared by Thomas
Simpson in 1755 (printed 1756) first applied the theory to the
discussion of errors of observation. The reprint (1757) of this
memoir lays down the axioms that positive and negative errors are
equally probable, and that there are certain assignable limits
within which all errors may be supposed to fall; continuous errors
are discussed and a probability curve is given.

Pierre-Simon
Laplace (1774) made the first attempt to deduce a rule for the
combination of observations from the principles of the theory of
probabilities. He represented the law of probability of errors by a
curve y = \phi(x), x being any error and y its probability, and
laid down three properties of this curve:

- it is symmetric as to the y-axis;
- the x-axis is an asymptote, the probability of the error \infty being 0;
- the area enclosed is 1, it being certain that an error exists.

The method
of least squares is due to Adrien-Marie
Legendre (1805), who introduced it in his Nouvelles méthodes
pour la détermination des orbites des comètes (New Methods for
Determining the Orbits of Comets). In ignorance of Legendre's
contribution, an Irish-American writer, Robert
Adrain, editor of "The Analyst" (1808), first deduced the law
of facility of error,

- \phi(x) = ce^,

h being a constant depending on precision of
observation, and c a scale factor ensuring that the area under the
curve equals 1. He gave two proofs, the second being essentially
the same as John
Herschel's (1850). Gauss
gave the first proof which seems to have been known in Europe (the
third after Adrain's) in 1809. Further proofs were given by Laplace
(1810, 1812), Gauss (1823),
James Ivory (1825, 1826), Hagen (1837), Friedrich
Bessel (1838), W. F.
Donkin (1844, 1856), and Morgan
Crofton (1870). Other contributors were Ellis (1844), De
Morgan (1864),
Glaisher (1872), and Giovanni
Schiaparelli (1875). Peters's (1856) formula for r, the
probable error of a single observation, is well known.

In the nineteenth
century authors on the general theory included Laplace, Sylvestre
Lacroix (1816), Littrow (1833), Adolphe
Quetelet (1853), Richard
Dedekind (1860), Helmert (1872), Hermann
Laurent (1873), Liagre, Didion, and Karl
Pearson. Augustus
De Morgan and George Boole
improved the exposition of the theory.

On the geometric side (see integral
geometry) contributors to The
Educational Times were influential (Miller, Crofton, McColl,
Wolstenholme, Watson, and Artemas Martin).

The opposite or complement of an event A is the
event [not A] (that is, the event of A not occurring); its
probability is given by . As an example, the chance of not rolling
a six on a six-sided die is = - \tfrac = \tfrac. See Complementary
event for a more complete treatment.

If two events, A and B are independent
then the joint
probability is

- P(A \mboxB) = P(A \cap B) = P(A) P(B),\,

If two events are mutually
exclusive then the probability of either occurring is

- P(A\mboxB) = P(A \cup B)= P(A) + P(B).

If the events are not mutually exclusive then

- \mathrm\left(A \hbox B\right)=\mathrm\left(A\right)+\mathrm\left(B\right)-\mathrm\left(A \mbox B\right).

Conditional
probability is the probability of some event A,
given the occurrence of some other event B. Conditional probability
is written P(A|B), and is read "the probability of A, given B". It
is defined by

- P(A \mid B) = \frac.\,

There have been at least two successful attempts
to formalize probability, namely the Kolmogorov
formulation and the Cox
formulation. In Kolmogorov's formulation (see probability
space), sets are
interpreted as events
and probability itself as a measure
on a class of sets. In Cox's
theorem, probability is taken as a primitive (that is, not
further analyzed) and the emphasis is on constructing a consistent
assignment of probability values to propositions. In both cases,
the laws of
probability are the same, except for technical details.

There are other methods for quantifying
uncertainty, such as the Dempster-Shafer
theory and possibility
theory, but those are essentially different and not compatible
with the laws of probability as they are usually understood.

A good example is the effect of the perceived
probability of any widespread Middle East conflict on oil prices -
which have ripple effects in the economy as a whole. An assessment
by a commodity trader that a war is more likely vs. less likely
sends prices up or down, and signals other traders of that opinion.
Accordingly, the probabilities are not assessed independently nor
necessarily very rationally. The theory of behavioral
finance emerged to describe the effect of such groupthink on pricing, on
policy, and on peace and conflict.

It can reasonably be said that the discovery of
rigorous methods to assess and combine probability assessments has
had a profound effect on modern society. Accordingly, it may be of
some importance to most citizens to understand how odds and
probability assessments are made, and how they contribute to
reputations and to decisions, especially in a democracy.

Another significant application of probability
theory in everyday life is
reliability. Many consumer products, such as automobiles and consumer
electronics, utilize reliability
theory in the design of the product in order to reduce the
probability of failure. The probability of failure is also closely
associated with the product's warranty.

A revolutionary discovery of 20th century physics
was the random character of all physical processes that occur at
microscopic scales and are governed by the laws of quantum
mechanics. The wave
function itself evolves deterministically as long as no
observation is made, but, according to the prevailing Copenhagen
interpretation, the randomness caused by the wave
function collapsing when an observation is made, is
fundamental. This means that probability
theory is required to describe nature. Others never came to
terms with the loss of determinism. Albert
Einstein famously
remarked in a letter to Max Born:
Jedenfalls bin ich überzeugt, daß der Alte nicht würfelt. (I am
convinced that God does not play dice). Although alternative
viewpoints exist, such as that of quantum
decoherence being the cause of an apparent random collapse, at
present there is a firm consensus among the physicists that
probability theory is necessary to describe quantum
phenomena.

- Decision theory
- Equiprobable
- Fuzzy measure theory
- Game theory
- Information theory
- Important publications in probability
- Measure theory
- Probabilistic argumentation
- Probabilistic logic
- Random fields
- Random variable
- Statistics
- List of statistical topics
- Stochastic process
- Wiener process
- Black Swan theory

- Olav Kallenberg, Probabilistic Symmetries and Invariance Principles. Springer -Verlag, New York (2005). 510 pp. ISBN 0-387-25115-4
- Kallenberg, O., Foundations of Modern Probability, 2nd ed. Springer Series in Statistics. (2002). 650 pp. ISBN 0-387-95313-2

- Damon Runyon, "It may be that the race is not always to the swift, nor the battle to the strong - but that is the way to bet."
- Pierre-Simon Laplace "It is remarkable that a science which began with the consideration of games of chance should have become the most important object of human knowledge." Théorie Analytique des Probabilités, 1812.
- Richard von Mises "The unlimited extension of the validity of the exact sciences was a characteristic feature of the exaggerated rationalism of the eighteenth century" (in reference to Laplace). Probability, Statistics, and Truth, p 9. Dover edition, 1981 (republication of second English edition, 1957).

- Edwin Thompson Jaynes. Probability Theory: The Logic of Science. Preprint: Washington University, (1996). — [http://omega.albany.edu:8008/JaynesBook.html HTML index with links to PostScript files] and PDF
- Dictionary of the History of Ideas: Certainty in Seventeenth-Century Thought
- Dictionary of the History of Ideas: Certainty since the Seventeenth Century
- Figures from the History of Probability and Statistics (Univ. of Southampton)
- Probability and Statistics on the Earliest Uses Pages (Univ. of Southampton)
- Earliest Uses of Symbols in Probability and Statistics on Earliest Uses of Various Mathematical Symbols
- A tutorial on probability and Bayes’ theorem devised for first-year Oxford University students

probabilities in Arabic: احتمال

probabilities in Bulgarian: Вероятност

probabilities in Bosnian: Vjerovatnoća

probabilities in Czech: Pravděpodobnost

probabilities in German:
Wahrscheinlichkeit

probabilities in Estonian: Tõenäosus

probabilities in Spanish: Probabilidad

probabilities in Esperanto: Probablo

probabilities in Persian: احتمالات

probabilities in French: Probabilité

probabilities in Korean: 확률

probabilities in Ido: Probableso

probabilities in Italian: Probabilità

probabilities in Hebrew: הסתברות

probabilities in Latvian: Varbūtība

probabilities in Dutch: Kans (statistiek)

probabilities in Japanese: 確率

probabilities in Norwegian: Sannsynlighet

probabilities in Polish:
Prawdopodobieństwo

probabilities in Portuguese: Probabilidade

probabilities in Romanian: Probabilitate

probabilities in Russian: Вероятность

probabilities in Simple English:
Probability

probabilities in Slovak: Pravdepodobnosť

probabilities in Serbian: Вероватноћа

probabilities in Sundanese: Probabilitas

probabilities in Finnish: Todennäköisyys

probabilities in Swedish: Sannolikhet

probabilities in Tamil: நிகழ்தகவு

probabilities in Thai: ความน่าจะเป็น

probabilities in Turkish: Olasılık

probabilities in Ukrainian: Ймовірність

probabilities in Chinese: 概率

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