Probabilitas: Béda antarrépisi
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Kecap '''''probabilitas''''' asalna tina basa [[Latin]] ''probare'' (ngabuktikeun, atawa nyoba).
Sacara teu resmi, ''probable''
''Chance'', ''odds'', jeung ''bet''
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==Historical remarks==
Probability theory, as applied to observations, was largely a [[nineteenth century]] development. [[Gambling]] shows that there has been an interest in quantifying the ideas of probability for millennia, but exact mathematical descriptions of use in these types of problems only arose much later.
The doctrine of probabilities dates as far back as [[Pierre de Fermat]] and [[Blaise Pascal]] (1654). [[Christiaan Huygens]] (1657) gave the first 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.
The theory of errors may be traced back to [[Roger Cotes]]'s ''Opera Miscellanea'' (posthumous, 1722), but a memoir prepared by 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.
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to the <math>y</math>-axis; (2) the <math>x</math>-axis is an asymptote, the probability of the error <math>\infty</math> being 0; (3) the area enclosed is 1, it being certain that an error exists. He deduced a formula for the mean of three observations. He also gave (1781) a formula for the law of facility of error (a term due to Lagrange, 1774), but one which led to unmanageable equations. Daniel Bernoulli (1778) introduced the principle of the maximum product of the probabilities of a system of concurrent errors.
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''. 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,
:<math>\phi(x) = ce^{-h^2 x^2}</math>
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1856), and Crofton (1870). Other contributors were Ellis (1844), [[De Morgan]] (1864), [[Glaisher]] (1872), and Schiaparelli (1875). Peters's (1856) formula for <math>r</math>, the probable error of a single observation, is well known.
In the [[nineteenth century]] authors on the general theory included Laplace, Lacroix (1816), Littrow (1833), [[Adolphe Quetelet]] (1853), [[Richard Dedekind]] (1860), Helmert (1872), Laurent (1873), Liagre, Didion, and 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).
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* [[Aleatory probability]], which represents the likelihood of future events whose occurrence is governed by some ''random'' physical phenomenon. This concept can be further divided into physical phenomena that are predictable, in principle, with sufficient information, and phenomena which are essentially unpredictable. Examples of the first kind include tossing [[dice]] or spinning a [[roulette]] wheel, and an example of the second kind is [[radioactive decay]].
* [[Epistemic probability]], which represents our uncertainty about propositions when one lacks complete knowledge of causative circumstances. Such propositions may be about past or future events, but need not be. Some examples of epistemic probability are to assign a probability to the proposition that a proposed law of physics is true, and to determine how "probable" it is that a suspect committed a crime, based on the evidence presented.
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The rather subtle distinction between "certain" and "probability 1" is treated at greater length in the article on "[[almost surely]]".
Most probabilities that occur in practice are numbers between 0 and 1, indicating the event's position on the continuum between impossibility and certainty. The closer an event's probability is to 1, the more likely it is to occur.
For example, if two events are assumed equally probable, such as a flipped coin landing heads-up or tails-up, we can express the probability of each event as "1 in 2", or, equivalently, "50%" or "1/2".
Probabilities are equivalently expressed as [[odds]], which is the ratio of the probability of one event to the probability of all other events.
The odds of heads-up, for the tossed coin, are (1/2)/(1 - 1/2), which is equal to 1/1. This is expressed as "1 to 1 odds" and often written "1:1".
Odds ''a'':''b'' for some event are equivalent to probability ''a''/(''a''+''b'').
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One approach is to use the [[law of large numbers]]. In this case, we assume that we can perform any number of coin flips, with each coin flip being independent - that is to say, the outcome of each coin flip is unaffected by previous coin flips. If we perform ''N'' trials (coin flips), and let ''N''<sub>H</sub> be the number of times the coin lands heads, then we can, for any ''N'', consider the ratio ''N''<sub>H</sub>/''N''.
As ''N'' gets larger and larger, we expect that in our example the ratio ''N''<sub>H</sub>/''N'' will get closer and closer to 1/2. This allows us to ''define'' the probability Pr(''H'') of flipping heads as the [[mathematical limit]], as ''N'' approaches infinity, of this sequence of ratios:
:<math>\Pr(H) = \lim_{N \to \infty}{N_H \over N} </math>
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The ''a priori'' aspect of this approach to probability is sometimes troubling when applied to real world situations. For example, in the play ''[[Rosencrantz and Guildenstern are Dead]]'' by [[Tom Stoppard]], a character flips a coin which keeps coming up heads over and over again, a hundred times. He can't decide whether this is just a random event - after all, it is possible (although unlikely) that a fair coin would give this result - or whether his assumption that the coin is fair is at fault.
=== Remarks on probability calculations ===
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To learn more about the basics of [[probability theory]], see the article on [[probability axiom]]s and the article on [[Bayes' theorem]] that explains the use of conditional probabilities in case where the occurrence of two events is related.
== Applications of probability theory to everyday life ==
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* [[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).
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