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| | == Mbt Shop Dublin down to $58.95 == |
| {{ProbabilityTopics}}
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| '''Probability''' is a measure or estimation of likelihood of occurrence of an event.<ref>[http://machaut.uchicago.edu/?resource=Webster%27s&word=probability&use1913=on "Probability"]. ''[[Webster's Dictionary|Webster’s Revised Unabridged Dictionary]]''. G & C Merriam, 1913</ref> Probabilities are given a value between 0 (0% chance or ''will not happen'') and 1 (100% chance or ''will happen'').<ref>[[William Feller|Feller, W.]] (1968), ''An Introduction to Probability Theory and its Applications'' (Volume 1). ISBN 0-471-25708-7 {{pn|date=June 2012}} </ref> The higher the degree of probability, the more likely the event is to happen, or, in a longer series of samples, the greater the number of times such event is expected to happen. A simple example is a [[coin toss]] that has 0.5 or 50% chance of landing with the "head" side facing up. | | Once you sift through the exterminators and make a decision, make sure they are using effective pesticides and bedbug control methods. Usually, exterminators will use a combination of pesticides and steam heat bed bugs only die from heat if the temperature is at least 120 degrees Fahrenheit that's sustained for several hours. Do not use a pest control company that utilizes bug foggers or bug bombs as their method for bed bugs; these methods fail to work. The only effective bed bug extermination methods for pesticides include direct contact.<br><br>I'm a visual communications student and also use her invitation as part of my portfolio. She is a great friend of mine, plus I want to expand my portfolio basically, won be charging her/making any income [http://www.gsar.ie/content/finder.asp Mbt Shop Dublin] off of them. Is it ok if I still use your font?<br><br>I had been fortunate enough tobenefit froman impressive increase in earningsduring Marchthat led to me having my best month since getting on the passive incomemerrygoround about Six months ago. It a small distinction, but I actually over the moon about it, because it indiciates that I not a onetrick pony that is totally dependent on AdSense for revenue. (Don get me wrong,I want as much AdSense revenue as possible, but I also want to be diversified in case I have a weak month regarding AdSense.) Revenue from WebAnswers decreased the very first time [http://www.liquidbar.co.nz/wp-content/plugins/formidable/classes/cache.php Isabel Marant Nz] (down to $58.95). Another first: WebAnswers was not the topearning site for me HubPages was. Again, these are things I happy about because it shows that the diversification of my eforts is paying off.<br><br>The damage was so bad that the county had to issue a situation of emergency that lasted for pretty much as week as it struggled to operate without phones, IT systems or internet for its employees. The Treasurer TMs Office, the Titles Office, even [http://www.brownsburgchatham.ca/bottin/feed.asp Beats Headphones Canada] the County Court and Prosecutor TMs office were left scrambling for their pens, notepads and carbon paper.<br><br>Woah my body system clock seriously have gone haywire. i didnt sleep the whole of yesterday night. closed my eyes limited to 12noon, yes 12noon. And woke up 3 hours after that hahaha mum thinks i ought to find a job that is in night shift hahaha since i am like an owl then slept back at 5pm and woke up almost at 10pm i think? Nowwwwwww i widely awake again hahahais i think i should get myself a container of sleeping pills soon. anyway all the best to all N [http://www.naturalseeding.co.nz/application/banner.html Louis Vuitton Auckland] level candidates for his or her results and have a great week ahead everyone xx<br><br>This is the first time that both Anna and Kejriwal came on the common platform since they parted ways in September this year. Kejriwal had, however, called on Hazare a number of times after the September 19 split. Anna, who'd split with Kejriwal on the question of anticorruption movement going for a political plunge, was on way to Amritsar where he will launch his nationwide tour on Sunday.<ul> |
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| These concepts have been given an axiomatic [[mathematics|mathematical]] derivation in [[probability theory]] (see [[probability axioms]]), which is used widely in such [[areas of study]] as [[mathematics]], [[statistics]], [[finance]], [[gambling]], [[science]], [[artificial intelligence]]/[[machine learning]] and [[philosophy]] to, for example, draw inferences about the expected frequency of events. Probability theory is also used to describe the underlying mechanics and regularities of [[complex systems]].<ref> [http://www.britannica.com/EBchecked/topic/477530/probability-theory Probability Theory] The Britannica website</ref>
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| ==Interpretations==
| | The Spaniards were attacked on the ocean. After one vessel was destroyed by a wellplaced shot, the others surrendered and produced about one million pounds in prize money. In 1995, i was incorporated as a notforprofit charitable organization, as well as in 1996 we moved to the former Glenburnie West School. A few years later, in 1999, we arranged to lease our current home the former St. <br><br>There is a two year service commitment when you signup and purchase your Video Phone device, one year for analog telephone adapter. All services, thereafter, are offered on a monthly basis. Hypersensitivity to the same acidic conditions can always exist. A comparison of rabeprazole was the recurrence of erosive esophagitis harm to TStrain mycoplasmas. <br><br>Come in for a visit to The Haunted Barn; don't get behind for the terror from using these barn walls is just waiting to feast on the warm flesh and blood of new prey. The creatures are climbing the walls seeking a getaway from The Haunted Barn but with the odor of new fresh warm blooded prey, they'll be coming after you. <br><br>This bothers many people, especially other unrecovered drunks who do not [http://www.grenvillesurlarouge.ca/Affichage/General/form.asp Mont Blanc Pens] even know him past his writings. But if they can deal with their own sensitivities, as I have learned, they will find my hubby to be a loving and generous man having a passion for sobriety like none we are able to imagine. <br><br>When I used to have to visit buy books for school as well as summer reading and we did not have it at the agency I called Barnes and Noble and Booksa Million [http://www.aldes.ie/flo/menu.asp Hollister Dublin] and much more often than not I would have had to order them. When I called Shaver's they more often than not had them in stock. <br><br>David Johnson opened the first Pita Pit in the city at 235 Third Avenue and even though he loves all his loyal customers (and they love his pitas) he keeps hoping Shania Twain will drop in on her next visit. [Shania, if you're reading this, call in your order to Dave at (705) 267PITA or fax it to him at (705) 2677483 and he'll have it waiting for you whenever you come by. <br><br>This all smacks of horrible complacency. Maybe on my small death bed I'll [http://www.risingstarspreschool.co.nz/images/new-page-images/layout.asp Nike Air Max 90] feel very different and be overwhelmed by these regrets. Watch your favourite programs in full screen at a time that suits you. Offering the largest selection of Old Time Radio [http://www.aldes.ie/flo/menu.asp Hollister Hoodies Ireland] downloads. <br><br>I understand a bunch of us have the 705 but cant recall that has the additional map software. I know there are cards w/ pre loaded data along with a dvd you can purchase as well. A free product enhances the transactional value of the acquisition. Use GWP offers in you spa and salon, for instance, with Every Hydrafacial treatments and receive a free Retinol Antiwrinkle Serum..<ul> |
| {{Main|Probability interpretations}}
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| | | <li>[http://profile.lcbc-leb.com/activity/p/102084/ http://profile.lcbc-leb.com/activity/p/102084/]</li> |
| When dealing with [[experiment]]s that are [[random]] and [[well-defined]] in a purely theoretical setting (like tossing a fair coin), probabilities describe the statistical number of outcomes considered divided by the number of all outcomes (tossing a fair coin twice will yield head-head with probability 1/4, because the four outcomes head-head, head-tails, tails-head and tails-tails are equally likely to occur). When it comes to practical application however there are two major conflicting categories of '''probability interpretations''', whose adherents possess different views about the fundamental nature of probability:
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| #[[Objectivists]] assign numbers to describe some objective or physical state of affairs. The most popular version of objective probability is [[frequentist probability]], which claims that the probability of a random event denotes the ''relative frequency of occurrence'' of an experiment's outcome, when repeating the experiment. This interpretation considers probability to be the relative frequency "in the long run" of outcomes.<ref>{{cite book |title=The Logic of Statistical Inference |first=Ian |last=Hacking |authorlink=Ian Hacking |year=1965 |publisher=Cambridge University Press|isbn=0-521-05165-7 }}{{pn|date=June 2012}}</ref> A modification of this is [[propensity probability]], which interprets probability as the tendency of some experiment to yield a certain outcome, even if it is performed only once.
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| #[[Subjective_probability#Objective_and_subjective_Bayesian_probabilities|Subjectivists]] assign numbers per subjective probability, i.e., as a degree of belief.<ref>{{cite journal |title=Logical foundations and measurement of subjective probability |first=Bruno de |last=Finetti |journal=Acta Psychologica |volume=34 |issue= |year=1970 |pages=129–145 |doi=10.1016/0001-6918(70)90012-0 }}</ref> The degree of belief has been interpreted as, "the price at which you would buy or sell a bet that pays 1 unit of utility if E, 0 if not E."<ref>{{cite web|last=Hájek|first=Alan|title=Interpretations of Probability|url=http://plato.stanford.edu/archives/win2012/entries/probability-interpret/|work=The Stanford Encyclopedia of Philosophy (Winter 2012 Edition), Edward N. Zalta (ed.)|accessdate=22 April 2013}}</ref> The most popular version of subjective probability is [[Bayesian probability]], which includes expert knowledge as well as experimental data to produce probabilities. The expert knowledge is represented by some (subjective) [[prior probability distribution]]. The data is incorporated in a likelihood function. The product of the prior and the likelihood, normalized, results in a [[posterior probability distribution]] that incorporates all the information known to date.<ref>{{cite book |title=Introduction to Mathematical Statistics |first=Robert V. |last=Hogg |first2=Allen |last2=Craig |first3=Joseph W. |last3=McKean |edition=6th |year=2004 |location=Upper Saddle River |publisher=Pearson |isbn=0-13-008507-3 }}{{pn|date=June 2012}}</ref> Starting from arbitrary, subjective probabilities for a group of agents, some Bayesians{{who|date=June 2012}} claim that all agents will eventually have sufficiently similar assessments of probabilities, given enough evidence (see [[Cromwell's rule]]).
| | <li>[http://site.vhostapp.com/news/html/?1908788.html http://site.vhostapp.com/news/html/?1908788.html]</li> |
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| ==Etymology==
| | <li>[http://www.xn--siqr3y30jp5c.cn/forum.php?mod=viewthread&tid=284081 http://www.xn--siqr3y30jp5c.cn/forum.php?mod=viewthread&tid=284081]</li> |
| The word ''Probability'' [[Derivation (linguistics)|derives]] from the Latin ''probabilitas'', which can also mean ''[[:wiktionary:probity|probity]]'', a measure of the [[authority]] of a [[witness]] in a [[legal case]] in [[Europe]], and often correlated with the witness's [[nobility]]. In a sense, this differs much from the modern meaning of ''probability'', which, in contrast, is a measure of the weight of [[empirical evidence]], and is arrived at from [[inductive reasoning]] and [[statistical inference]].<ref name=Emergence>[[Ian Hacking|Hacking, I.]] (2006) ''The Emergence of Probability: A Philosophical Study of Early Ideas about Probability, Induction and Statistical Inference'', Cambridge University Press, ISBN 978-0-521-68557-3 {{pn|date=June 2012}}</ref>
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| ==History==
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| {{Main|History of probability}}
| | </ul> |
| 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 arose much later. There are reasons of course, for the slow development of the mathematics of probability. Whereas games of chance provided the impetus for the mathematical study of probability, fundamental issues are still obscured by the superstitions of gamblers.<ref>Freund, John. (1973) ''Introduction to Probability''. Dickenson ISBN 978-0822100782 (p. 1)</ref>
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| [[File:Christiaan Huygens-painting.jpeg|thumb|140px|Christiaan Huygens probably published the first book on probability]]
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| 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."<ref name="Jeffrey">Jeffrey, R.C., ''Probability and the Art of Judgment,'' Cambridge University Press. (1992). pp. 54-55 . ISBN 0-521-39459-7</ref> However, in legal contexts especially, 'probable' could also apply to propositions for which there was good evidence.<ref name="Franklin">Franklin, J. (2001) ''The Science of Conjecture: Evidence and Probability Before Pascal,'' Johns Hopkins University Press. (pp. 22, 113, 127)</ref>
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| Aside from elementary work by [[Gerolamo 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.<ref>{{citation|url=http://www.secondmoment.org/articles/probability.php|publisher=Second Moment|accessdate=2008-05-23|title=A Brief History of Probability|last=Abrams|first=William}}</ref> [[Jakob Bernoulli|Jakob Bernoulli's]] ''[[Ars Conjectandi]]'' (posthumous, 1713) and [[Abraham de Moivre|Abraham de Moivre's]] ''[[Doctrine of Chances]]'' (1718) treated the subject as a branch of mathematics.<ref>{{Cite book | last1 = Ivancevic | first1 = Vladimir G.
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| | last2 = Ivancevic | first2 = Tijana T.
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| | title = Quantum leap : from Dirac and Feynman, across the universe, to human body and mind
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| | year = 2008 | publisher = World Scientific
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| | location = Singapore ; Hackensack, NJ | isbn = 978-981-281-927-7
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| | page = 16 }}</ref> See [[Ian Hacking|Ian Hacking's]] ''The Emergence of Probability''<ref name=Emergence/> and [[James Franklin (philosopher)|James Franklin's]] ''The Science of Conjecture''{{full|date=November 2012}} for histories of the early development of the very concept of mathematical probability.
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| The [[theory of errors]] may be traced back to [[Roger Cotes|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.{{Citation needed|date=February 2012}} The reprint (1757) of this memoir lays down the axioms that positive and negative errors are equally probable, and that certain assignable limits define the range of all errors. Simpson also discusses continuous errors and describes a probability curve.
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| The first two laws of error that were proposed both originated with [[Pierre-Simon Laplace]]. The first law was published in 1774 and stated that the frequency of an error could be expressed as an exponential function of the numerical magnitude of the error, disregarding sign. The second law of error was proposed in 1778 by Laplace and stated that the frequency of the error is an exponential function of the square of the error.<ref name=Wilson1923>Wilson EB (1923) "First and second laws of error". [[Journal of the American Statistical Association]], 18, 143</ref> The second law of error is called the normal distribution or the Gauss law. "It is difficult historically to attribute that law to Gauss, who in spite of his well-known precocity had probably not made this discovery before he was two years old."<ref name=Wilson1923 />
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| [[Daniel Bernoulli]] (1778) introduced the principle of the maximum product of the probabilities of a system of concurrent errors.
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| [[File:Bendixen - Carl Friedrich Gauß, 1828.jpg|thumb|140px|Carl Friedrich Gauss]]
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| [[Adrien-Marie Legendre]] (1805) developed the [[method of least squares]], and introduced it in his ''Nouvelles méthodes pour la détermination des orbites des comètes'' (''New Methods for Determining the Orbits of Comets'').{{cn|date=June 2012}} 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,
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| :<math>\phi(x) = ce^{-h^2 x^2},</math>
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| where <math>h</math> is a constant depending on precision of observation, and <math>c</math> is 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|John Herschel's]] (1850).{{cn|date=June 2012}} [[Carl Friedrich Gauss|Gauss]] gave the first proof that 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 (mathematician)|James Ivory]] (1825, 1826), Hagen (1837), [[Friedrich Bessel]] (1838), [[W. F. Donkin]] (1844, 1856), and [[Morgan Crofton]] (1870). Other contributors were Ellis (1844), [[Augustus De Morgan|De Morgan]] (1864), [[James Whitbread Lee Glaisher|Glaisher]] (1872), and [[Giovanni Schiaparelli]] (1875). [[Christian August Friedrich Peters|Peters]]'s (1856) formula{{clarify|date=June 2012}} for ''r'', the [[probable error]] of a single observation, is well known.{{to whom|date=May 2012}}
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| 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.
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| [[Andrey Markov]] introduced{{Citation needed|date=February 2012}} the notion of [[Markov chains]] (1906), which played an important role in [[stochastic process]]es theory and its applications. The modern theory of probability based on the [[Measure (mathematics)|measure theory]] was developed by [[Andrey Kolmogorov]] (1931).{{cn|date=June 2012}}
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| On the geometric side (see [[integral geometry]]) contributors to ''[[The Educational Times]]'' were influential (Miller, Crofton, McColl, Wolstenholme, Watson, and [[Artemas Martin]]).{{Citation needed|date=February 2012}}
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| {{Further|History of statistics}}
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| ==Theory==
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| {{Main|Probability theory}}
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| Like other [[theory|theories]], the [[probability theory|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 interpreted or translated back into the problem domain.
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| There have been at least two successful attempts to formalize probability, namely the [[Kolmogorov]] formulation and the [[Richard Threlkeld Cox|Cox]] formulation. In Kolmogorov's formulation (see [[probability space]]), [[Set (mathematics)|sets]] are interpreted as [[Event (probability theory)|event]]s and probability itself as a [[Measure (mathematics)|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 [[probability axioms|laws of probability]] are the same, except for technical details.
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| There are other methods for quantifying uncertainty, such as the [[Dempster–Shafer theory]] or [[possibility theory]], but those are essentially different and not compatible with the laws of probability as usually understood.
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| == Applications ==
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| Probability theory is applied in everyday life in [[risk]] assessment and in trade on [[financial market]]s. Governments apply probabilistic methods in [[environmental regulation]], where it is called pathway analysis.
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| 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 neither 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.<ref>Singh, Laurie (2010) "Whither Efficient Markets? Efficient Market Theory and Behavioral Finance". The Finance Professionals' Post, 2010.</ref>
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| The discovery of rigorous methods to assess and combine probability assessments has changed society. It is important for most citizens to understand how probability assessments are made, and how they contribute to decisions.
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| Another significant application of probability theory in everyday life is [[Reliability theory of aging and longevity|reliability]]. Many consumer products, such as [[automobiles]] and consumer electronics, use [[reliability theory]] in product design to reduce the probability of failure. Failure probability may influence a manufacturer's decisions on a product's [[warranty]].<ref>Gorman, Michael (2011) "Management Insights". ''Management Science'' {{full|date=November 2012}}</ref>
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| The [[cache language model]] and other [[Statistical Language Model|statistical language models]] that are used in [[natural language processing]] are also examples of applications of probability theory.
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| ==Mathematical treatment==
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| {{see also|Probability axioms}}
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| Consider an experiment that can produce a number of results. The collection of all results is called the sample space of the experiment. The [[power set]] of the sample space is formed by considering all different collections of possible results. For example, rolling a die can produce six possible results. One collection of possible results gives an odd number on the die. Thus, the subset {1,3,5} is an element of the [[power set]] of the sample space of die rolls. These collections are called "events." In this case, {1,3,5} is the event that the die falls on some odd number. If the results that actually occur fall in a given event, the event is said to have occurred.
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| A probability is a [[Function (mathematics)|way of assigning]] every event a value between zero and one, with the requirement that the event made up of all possible results (in our example, the event {1,2,3,4,5,6}) is assigned a value of one. To qualify as a probability, the assignment of values must satisfy the requirement that if you look at a collection of mutually exclusive events (events with no common results, e.g., the events {1,6}, {3}, and {2,4} are all mutually exclusive), the probability that at least one of the events will occur is given by the sum of the probabilities of all the individual events.<ref>Ross, Sheldon. ''A First course in Probability'', 8th Edition. Page 26-27.</ref>
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| The probability of an [[Event (probability theory)|event]] ''A'' is written as ''P''(''A''), ''p''(''A'') or Pr(''A'').<ref>Olofsson (2005) Page 8.</ref> This mathematical definition of probability can extend to infinite sample spaces, and even uncountable sample spaces, using the concept of a measure.
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| 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 {{nowrap|1= ''P''(not ''A'') = 1 − ''P''(''A'')}}.<ref>Olofsson (2005), page 9</ref> As an example, the chance of not rolling a six on a six-sided die is {{nowrap|1 – (chance of rolling a six) }}<math>= 1 - \tfrac{1}{6} = \tfrac{5}{6}</math>. See [[Complementary event]] for a more complete treatment.
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| If two events ''A'' and ''B'' occur on a single performance of an experiment, this is called the intersection or [[Joint distribution|joint probability]] of ''A'' and ''B'', denoted as <math>P(A \cap B)</math>.
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| ===Independent events===
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| If two events, ''A'' and ''B'' are [[Independence (probability theory)|independent]] then the joint probability is
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| :<math>P(A \mbox{ and }B) = P(A \cap B) = P(A) P(B),\,</math>
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| for example, if two coins are flipped the chance of both being heads is <math>\tfrac{1}{2}\times\tfrac{1}{2} = \tfrac{1}{4}.</math><ref>Olofsson (2005) page 35.</ref>
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| ====Mutually exclusive events====
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| If either event ''A'' or event ''B'' or both events occur on a single performance of an experiment this is called the union of the events ''A'' and ''B'' denoted as <math>P(A \cup B)</math>.
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| If two events are [[Mutually exclusive events|mutually exclusive]] then the probability of either occurring is
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| :<math>P(A\mbox{ or }B) = P(A \cup B)= P(A) + P(B).</math>
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| For example, the chance of rolling a 1 or 2 on a six-sided {{dice}} is <math>P(1\mbox{ or }2) = P(1) + P(2) = \tfrac{1}{6} + \tfrac{1}{6} = \tfrac{1}{3}.</math>
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| ====Not mutually exclusive events====
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| If the events are not mutually exclusive then
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| :<math>P\left(A \hbox{ or } B\right)=P\left(A\right)+P\left(B\right)-P\left(A \mbox{ and } B\right).</math>
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| For example, when drawing a single card at random from a regular deck of cards, the chance of getting a heart or a face card (J,Q,K) (or one that is both) is <math>\tfrac{13}{52} + \tfrac{12}{52} - \tfrac{3}{52} = \tfrac{11}{26}</math>, because of the 52 cards of a deck 13 are hearts, 12 are face cards, and 3 are both: here the possibilities included in the "3 that are both" are included in each of the "13 hearts" and the "12 face cards" but should only be counted once.
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| ===Conditional probability===
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| ''[[Conditional probability]]'' is the probability of some event ''A'', given the occurrence of some other event ''B''.
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| Conditional probability is written <math>P(A \mid B)</math>, and is read "the probability of ''A'', given ''B''". It is defined by<ref>Olofsson (2005) page 29.</ref>
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| :<math>P(A \mid B) = \frac{P(A \cap B)}{P(B)}.\,</math>
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| If <math>P(B)=0</math> then <math>P(A \mid B)</math> is formally [[defined and undefined|undefined]] by this expression. However, it is possible to define a conditional probability for some zero-probability events using a [[σ-algebra]] of such events (such as those arising from a [[continuous random variable]]).{{cn|date=July 2012}}
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| For example, in a bag of 2 red balls and 2 blue balls (4 balls in total), the probability of taking a red ball is <math>1/2</math>; however, when taking a second ball, the probability of it being either a red ball or a blue ball depends on the ball previously taken, such as, if a red ball was taken, the probability of picking a red ball again would be <math>1/3</math> since only 1 red and 2 blue balls would have been remaining.
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| ===Inverse probability===
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| In [[probability theory]] and applications, '''Bayes' rule''' relates the [[odds]] of event <math>A_1</math> to event <math>A_2</math>, before (prior to) and after (posterior to) [[Conditional probability|conditioning]] on another event <math>B</math>. The odds on <math>A_1</math> to event <math>A_2</math> is simply the ratio of the probabilities of the two events. When arbitrarily many events <math>A</math> are of interest, not just two, the rule can be rephrased as '''posterior is proportional to prior times likelihood''', <math>P(A|B)\propto P(A) P(B|A)</math> where the proportionality symbol means that the left hand side is proportional to (i.e., equals a constant times) the right hand side as <math>A</math> varies, for fixed or given <math>B</math> (Lee, 2012; Bertsch McGrayne, 2012). In this form it goes back to Laplace (1774) and to Cournot (1843); see Fienberg (2005). See [[Inverse probability]] and [[Bayes' rule]].
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| ===Summary of probabilities===
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| {| class="wikitable"
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| |+Summary of probabilities
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| |-
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| !Event!!Probability
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| |-
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| |align=center|A||<math>P(A)\in[0,1]\,</math>
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| |-
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| |align=center|not A||<math>P(A^c)=1-P(A)\,</math>
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| |-
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| |align=center|A or B
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| |<math>\begin{align}
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| P(A\cup B) & = P(A)+P(B)-P(A\cap B) \\
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| P(A\cup B) & = P(A)+P(B) \qquad\mbox{if A and B are mutually exclusive} \\
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| \end{align}</math>
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| |-
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| |align=center|A and B
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| |<math>\begin{align}
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| P(A\cap B) & = P(A|B)P(B) = P(B|A)P(A)\\
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| P(A\cap B) & = P(A)P(B) \qquad\mbox{if A and B are independent}\\
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| \end{align}</math>
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| |-
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| |align=center|A given B
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| |<math>P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{P(B|A)P(A)}{P(B)} \,</math>
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| |}
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| == Relation to randomness ==
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| {{Main|Randomness}}
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| In a [[determinism|deterministic]] universe, based on [[Newtonian mechanics|Newtonian]] concepts, there would be no probability if all conditions were known ([[Laplace's demon]]), (but there are situations in which [[chaos theory|sensitivity to initial conditions]] exceeds our ability to measure them, i.e. know them). In the case of a roulette wheel, if the force of the hand and the period of that force are known, the number on which the ball will stop would be a certainty (though as a practical matter, this would likely be true only of a roulette wheel that had not been exactly levelled — as Thomas A. Bass' [[eudaemons|Newtonian Casino]] revealed). Of course, this also assumes knowledge of inertia and friction of the wheel, weight, smoothness and roundness of the ball, variations in hand speed during the turning and so forth. A probabilistic description can thus be more useful than Newtonian mechanics for analyzing the pattern of outcomes of repeated rolls of a roulette wheel. Physicists face the same situation in [[kinetic theory]] of gases, where the system, while deterministic ''in principle'', is so complex (with the number of molecules typically the order of magnitude of [[Avogadro constant]] 6.02·10<sup>23</sup>) that only a statistical description of its properties is feasible.
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| [[Probability theory]] is required to describe quantum phenomena.<ref>Burgi, Mark (2010) "Interpretations of Negative Probabilities", p. 1. {{arXiv|1008.1287v1}}</ref> A revolutionary discovery of early 20th century [[physics]] was the random character of all physical processes that occur at sub-atomic scales and are governed by the laws of [[quantum mechanics]]. The objective [[wave function]] evolves deterministically but, according to the [[Copenhagen interpretation]], it deals with probabilities of observing, the outcome being explained by a [[wave function collapse]] when an observation is made. However, the loss of [[determinism]] for the sake of [[instrumentalism]] did not meet with universal approval. [[Albert Einstein]] famously [[:de:Albert Einstein#Quellenangaben und Anmerkungen|remarked]] in a letter to [[Max Born]]: "I am convinced that God does not play dice".<ref>''Jedenfalls bin ich überzeugt, daß der Alte nicht würfelt.'' Letter to Max Born, 4 December 1926, in: [http://books.google.de/books?id=LQIsAQAAIAAJ&q=achtung-gebietend Einstein/Born Briefwechsel 1916-1955].</ref> Like Einstein, [[Erwin Schrödinger]], who [[Schrödinger equation#Historical background and development|discovered]] the wave function, believed quantum mechanics is a [[statistical]] approximation of an underlying deterministic reality.<ref>{{cite book |last=Moore |first=W.J. |year=1992 |title=Schrödinger: Life and Thought |publisher=[[Cambridge University Press]] |page=479 |isbn=0-521-43767-9}}</ref> In modern interpretations, [[quantum decoherence]] accounts for subjectively probabilistic behavior.
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| == See also ==
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| {{Portal|Logic}}
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| {{Main|Outline of probability}}
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| * [[Chance (disambiguation)]]
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| * [[Class membership probabilities]]
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| * [[Equiprobability]]
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| * [[Heuristics in judgment and decision making]]
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| == Notes ==
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| {{Reflist|30em}}
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| ==Bibliography==
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| * [[Olav Kallenberg|Kallenberg, O.]] (2005) ''Probabilistic Symmetries and Invariance Principles''. Springer -Verlag, New York. 510 pp. ISBN 0-387-25115-4
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| * Kallenberg, O. (2002) ''Foundations of Modern Probability,'' 2nd ed. Springer Series in Statistics. 650 pp. ISBN 0-387-95313-2
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| *Olofsson, Peter (2005) ''Probability, Statistics, and Stochastic Processes'', Wiley-Interscience. 504 pp ISBN 0-471-67969-0.
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| ==External links==
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| {{Wikiquote}}
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| {{Wikibooks|Probability}}
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| {{Library resources box
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| |by=no
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| |onlinebooks=no
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| |others=no
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| |about=yes
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| |label=Probability}}
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| * [http://www.math.uah.edu/stat/ Virtual Laboratories in Probability and Statistics (Univ. of Ala.-Huntsville)]
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| * {{In Our Time|Probability|b00bqf61|Probability}}
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| *[http://wiki.stat.ucla.edu/socr/index.php/EBook Probability and Statistics EBook]
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| *[[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 [http://bayes.wustl.edu/etj/prob/book.pdf PDF] (first three chapters)
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| *[http://www.economics.soton.ac.uk/staff/aldrich/Figures.htm People from the History of Probability and Statistics (Univ. of Southampton)]
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| *[http://www.economics.soton.ac.uk/staff/aldrich/Probability%20Earliest%20Uses.htm Probability and Statistics on the Earliest Uses Pages (Univ. of Southampton)]
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| *[http://jeff560.tripod.com/stat.html Earliest Uses of Symbols in Probability and Statistics] on [http://jeff560.tripod.com/mathsym.html Earliest Uses of Various Mathematical Symbols]
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| *[http://www.celiagreen.com/charlesmccreery/statistics/bayestutorial.pdf A tutorial on probability and Bayes’ theorem devised for first-year Oxford University students]
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| *[http://ubu.com/historical/young/index.html] pdf file of [[An Anthology of Chance Operations]] (1963)] at [[UbuWeb]]
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| *[http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html Introduction to Probability - eBook], by Charles Grinstead, Laurie Snell [http://bitbucket.org/shabbychef/numas_text/ Source] ''([[GNU Free Documentation License]])''
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| * {{en icon}} {{it icon}} [[Bruno de Finetti]], ''[http://amshistorica.unibo.it/35 Probabilità e induzione]'', Bologna, CLUEB, 1993. ISBN 88-8091-176-7 (digital version)
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