The probabilities of rolling several numbers using two dice
Probability is a branch ofmathematics andstatistics concerningevents and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur.[note 1][1][2] This number is often expressed as a percentage (%), ranging from 0% to 100%. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%).
When dealing withrandom experiments – i.e.,experiments that arerandom andwell-defined – in a purely theoretical setting (like tossing a coin), probabilities can be numerically described by the number of desired outcomes, divided by the total number of all outcomes. This is referred to astheoretical probability (in contrast toempirical probability, dealing with probabilities in the context of real experiments). The probability is a number between 0 and 1; the larger the probability, the more likely the desired outcome is to occur. For example, tossing a coin twice will yield "head-head", "head-tail", "tail-head", and "tail-tail" outcomes. The probability of getting an outcome of "head-head" is 1 out of 4 outcomes, or, in numerical terms, 1/4, 0.25 or 25%. The probability of getting an outcome of at least one head is 3 out of 4, or 0.75, and this event is more likely to occur. However, when it comes to practical application, there are two major competing categories of probability interpretations, whose adherents hold different views about the fundamental nature of probability:
Objectivists assign numbers to describe some objective or physical state of affairs. The most popular version of objective probability isfrequentist probability, which claims that the probability of a random event denotes therelative frequency of occurrence of an experiment's outcome when the experiment is repeated indefinitely. This interpretation considers probability to be the relative frequency "in the long run" of outcomes.[5] A modification of this ispropensity probability, which interprets probability as the tendency of some experiment to yield a certain outcome, even if it is performed only once.
Subjectivists assign numbers per subjective probability, that is, as adegree of belief.[6] 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",[7] although that interpretation is not universally agreed upon.[8] The most popular version of subjective probability isBayesian probability, which includes expert knowledge as well as experimental data to produce probabilities. The expert knowledge is represented by some (subjective)prior probability distribution. These data are incorporated in alikelihood function. The product of the prior and the likelihood, when normalized, results in aposterior probability distribution that incorporates all the information known to date.[9] ByAumann's agreement theorem, Bayesian agents whose prior beliefs are similar will end up with similar posterior beliefs. However, sufficiently different priors can lead to different conclusions, regardless of how much information the agents share.[10]
The scientific study of probability is a modern development of mathematics.Gambling shows that there has been an interest in quantifying the ideas of probability throughout history, but exact mathematical descriptions arose much later. There are reasons for the slow development of the mathematics of probability. Whereas games of chance provided the impetus for the mathematical study of probability, fundamental issues[note 2] are still obscured by superstitions.[11]
According toRichard Jeffrey, "Before the middle of the seventeenth century, the term 'probable' (Latinprobabilis) meantapprovable, 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."[12] However, in legal contexts especially, 'probable' could also apply to propositions for which there was good evidence.[13]
The sixteenth-centuryItalian polymathGerolamo Cardano demonstrated the efficacy of definingodds as the ratio of favourable to unfavourable outcomes (which implies that the probability of an event is given by the ratio of favourable outcomes to the total number of possible outcomes[14]).Aside from the elementary work by Cardano, the doctrine of probabilities dates to the correspondence ofPierre de Fermat andBlaise Pascal (1654).Christiaan Huygens (1657) gave the earliest known scientific treatment of the subject.[15]Jakob Bernoulli'sArs Conjectandi (posthumous, 1713) andAbraham de Moivre'sDoctrine of Chances (1718) treated the subject as a branch of mathematics.[16] SeeIan Hacking'sThe Emergence of Probability[4] andJames Franklin'sThe Science of Conjecture[17] for histories of the early development of the very concept of mathematical probability.
Thetheory of errors may be traced back toRoger Cotes'sOpera Miscellanea (posthumous, 1722), but a memoir prepared byThomas Simpson in 1755 (printed 1756) first applied the theory to the discussion of errors of observation.[18] 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.
The first two laws of error that were proposed both originated withPierre-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.[19] 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."[19]
Daniel Bernoulli (1778) introduced the principle of the maximum product of the probabilities of a system of concurrent errors.
Carl Friedrich Gauss
Adrien-Marie Legendre (1805) developed themethod of least squares, and introduced it in hisNouvelles méthodes pour la détermination des orbites des comètes (New Methods for Determining the Orbits of Comets).[20] 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,
where is a constant depending on precision of observation, and is a scale factor ensuring that the area under the curve equals 1. He gave two proofs, the second being essentially the same asJohn Herschel's (1850).[citation needed]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 (1825, 1826), Hagen (1837),Friedrich Bessel (1838),W.F. Donkin (1844, 1856), andMorgan Crofton (1870). Other contributors wereEllis (1844),De Morgan (1864),Glaisher (1872), andGiovanni Schiaparelli (1875).Peters's (1856) formula[clarification needed] forr, theprobable error of a single observation, is well known.
On the geometric side, contributors toThe Educational Times included Miller, Crofton, McColl, Wolstenholme, Watson, andArtemas Martin.[23] Seeintegral geometry for more information.
Like othertheories, thetheory of probability is a representation of its 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.
There have been at least two successful attempts to formalize probability, namely theKolmogorov formulation and theCox formulation. In Kolmogorov's formulation (see alsoprobability space),sets are interpreted asevents and probability as ameasure on a class of sets. InCox's theorem, probability is taken as a primitive (i.e., not further analyzed), and the emphasis is on constructing a consistent assignment of probability values to propositions. In both cases, thelaws of probability are the same, except for technical details.
There are other methods for quantifying uncertainty, such as theDempster–Shafer theory orpossibility theory, but those are essentially different and not compatible with the usually-understood laws of probability.
An example of the use of probability theory in equity trading 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 can send that commodity's prices up or down, and signals other traders of that opinion. Accordingly, the probabilities are neither assessed independently nor necessarily rationally. The theory ofbehavioral finance emerged to describe the effect of suchgroupthink on pricing, on policy, and on peace and conflict.[24]
In addition to financial assessment, probability can be used to analyze trends in biology (e.g., disease spread) as well as ecology (e.g., biologicalPunnett squares).[25] As with finance, risk assessment can be used as a statistical tool to calculate the likelihood of undesirable events occurring, and can assist with implementing protocols to avoid encountering such circumstances. Probability is used to designgames of chance so that casinos can make a guaranteed profit, yet provide payouts to players that are frequent enough to encourage continued play.[26]
Another significant application of probability theory in everyday life isreliability. Many consumer products, such asautomobiles 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'swarranty.[27]
Consider an experiment that can produce a number of results. The collection of all possible results is called thesample space of the experiment, sometimes denoted as. Thepower 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 thepower set of the sample space of dice 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.
A probability is away 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 for any collection of mutually exclusive events (events with no common results, such as the events {1,6}, {3}, and {2,4}), the probability that at least one of the events will occur is given by the sum of the probabilities of all the individual events.[28]
The probability of aneventA is written as,[29], or.[30] This mathematical definition of probability can extend to infinite sample spaces, and even uncountable sample spaces, using the concept of a measure.
Theopposite orcomplement of an eventA is the event [notA] (that is, the event ofA not occurring), often denoted as,, or; its probability is given byP(notA) = 1 −P(A).[31] As an example, the chance of not rolling a six on a six-sided die is1 – (chance of rolling a six) =1 −1/6 =5/6. For a more comprehensive treatment, seeComplementary event.
If two eventsA andB occur on a single performance of an experiment, this is called the intersection orjoint probability ofA andB, denoted as
If either eventA or eventB can occur but never both simultaneously, then they are called mutually exclusive events.
If two events aremutually exclusive, then the probability ofboth occurring is denoted as and If two events aremutually exclusive, then the probability ofeither occurring is denoted as and
For example, the chance of rolling a 1 or 2 on a six-sided die is
If the events are not (necessarily) mutually exclusive then Rewritten,
For example, when drawing a card from a deck of cards, the chance of getting a heart or a face card (J, Q, K) (or both) is since among 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.
This can be expanded further for multiple not (necessarily) mutually exclusive events. For three events, this proceeds as follows:It can be seen, then, that this pattern can be repeated for any number of events.
Conditional probability is the probability of some eventA, given the occurrence of some other eventB. Conditional probability is written, and is read "the probability ofA, givenB". It is defined by[33]
If then is formallyundefined by this expression. In this case and are independent, since However, it is possible to define a conditional probability for some zero-probability events, for example by using aσ-algebra of such events (such as those arising from acontinuous random variable).[34]
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 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. For example, if a red ball was taken, then the probability of picking a red ball again would be since only 1 red and 2 blue balls would have been remaining. And if a blue ball was taken previously, the probability of taking a red ball will be
Inprobability theory and applications,Bayes' rule relates theodds of event to event before (prior to) and after (posterior to)conditioning on another event The odds on to event is simply the ratio of the probabilities of the two events. When arbitrarily many events are of interest, not just two, the rule can be rephrased asposterior is proportional to prior times likelihood, where the proportionality symbol means that the left hand side is proportional to (i.e., equals a constant times) the right hand side as varies, for fixed or given (Lee, 2012; Bertsch McGrayne, 2012). In this form it goes back to Laplace (1774) and to Cournot (1843); see Fienberg (2005).
In adeterministic universe, based onNewtonian concepts, there would be no probability if all conditions were known (Laplace's demon) (but there are situations in whichsensitivity to initial conditions exceeds our ability to measure them, i.e. know them). In the case of aroulette 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'Newtonian Casino revealed). 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 thekinetic theory of gases, where the system, while deterministicin principle, is so complex (with the number of molecules typically the order of magnitude of theAvogadro constant6.02×1023) that only a statistical description of its properties is feasible.[35]
Probability theory is required to describe quantum phenomena.[36] A revolutionary discovery of early 20th centuryphysics was the random character of all physical processes that occur at sub-atomic scales and are governed by the laws ofquantum mechanics. The objectivewave function evolves deterministically but, according to theCopenhagen interpretation, it deals with probabilities of observing, the outcome being explained by awave function collapse when an observation is made. However, the loss ofdeterminism for the sake ofinstrumentalism did not meet with universal approval.Albert Einstein famouslyremarked in a letter toMax Born: "I am convinced that God does not play dice".[37] Like Einstein,Erwin Schrödinger, whodiscovered the wave function, believed quantum mechanics is astatistical approximation of an underlying deterministicreality.[38] In some modern interpretations of the statistical mechanics of measurement,quantum decoherence is invoked to account for the appearance of subjectively probabilistic experimental outcomes.
^Strictly speaking, a probability of 0 indicates that an eventalmost never takes place, whereas a probability of 1 indicates than an eventalmost certainly takes place. This is an important distinction when thesample space is infinite. For example, for thecontinuous uniform distribution on thereal interval [5, 10], there are an infinite number of possible outcomes, and the probability of any given outcome being observed – for instance, exactly 7 – is 0. This means that an observation willalmost surely not be exactly 7. However, it doesnot mean that exactly 7 isimpossible. Ultimately some specific outcome (with probability 0) will be observed, and one possibility for that specific outcome is exactly 7.
^In the context of the book that this is quoted from, it is the theory of probability and the logic behind it that governs the phenomena of such things compared to rash predictions that rely on pure luck or mythological arguments such as gods of luck helping the winner of the game.
^abHacking, I. (2006)The Emergence of Probability: A Philosophical Study of Early Ideas about Probability, Induction and Statistical Inference, Cambridge University Press,ISBN978-0-521-68557-3[page needed]
^Finetti, Bruno de (1970). "Logical foundations and measurement of subjective probability".Acta Psychologica.34:129–145.doi:10.1016/0001-6918(70)90012-0.
^Hájek, Alan (21 October 2002). Edward N. Zalta (ed.)."Interpretations of Probability".The Stanford Encyclopedia of Philosophy (Winter 2012 ed.). Retrieved22 April 2013.
^Jaynes, E.T. (2003). "Section A.2 The de Finetti system of probability". In Bretthorst, G. Larry (ed.).Probability Theory: The Logic of Science (1 ed.). Cambridge University Press.ISBN978-0-521-59271-0.
^Hogg, Robert V.; Craig, Allen; McKean, Joseph W. (2004).Introduction to Mathematical Statistics (6th ed.). Upper Saddle River: Pearson.ISBN978-0-13-008507-8.[page needed]
^Jaynes, E.T. (2003). "Section 5.3 Converging and diverging views". In Bretthorst, G. Larry (ed.).Probability Theory: The Logic of Science (1 ed.). Cambridge University Press.ISBN978-0-521-59271-0.
^Ivancevic, Vladimir G.; Ivancevic, Tijana T. (2008).Quantum leap : from Dirac and Feynman, across the universe, to human body and mind. Singapore; Hackensack, NJ: World Scientific. p. 16.ISBN978-981-281-927-7.
^Franklin, James (2001).The Science of Conjecture: Evidence and Probability Before Pascal. Johns Hopkins University Press.ISBN978-0-8018-6569-5.
^Seneta, Eugene William.""Adrien-Marie Legendre" (version 9)".StatProb: The Encyclopedia Sponsored by Statistics and Probability Societies. Archived fromthe original on 3 February 2016. Retrieved27 January 2016.
^Weber, Richard."Markov Chains"(PDF).Statistical Laboratory. University of Cambridge.
^Wilcox, Rand R. (2016).Understanding and applying basic statistical methods using R. Hoboken, New Jersey.ISBN978-1-119-06140-3.OCLC949759319.{{cite book}}: CS1 maint: location missing publisher (link)
^Singh, Laurie (2010) "Whither Efficient Markets? Efficient Market Theory and Behavioral Finance". The Finance Professionals' Post, 2010.
^Edwards, Anthony William Fairbank (September 2012)."Reginald Crundall Punnett: First Arthur Balfour Professor of Genetics, Cambridge, 1912". Perspectives.Genetics.192 (1). Gonville and Caius College, Cambridge, UK:Genetics Society of America:3–13.doi:10.1534/genetics.112.143552.PMC3430543.PMID22964834. pp. 5–6:[…] Punnett's square seems to have been a development of 1905, too late for the first edition of hisMendelism (May 1905) but much in evidence inReport III to the Evolution Committee of the Royal Society [(Bateson et al. 1906b) "received March 16, 1906"]. The earliest mention is contained in a letter to Bateson from Francis Galton dated October 1, 1905 (Edwards 2012). We have the testimony of Bateson (1909, p. 57) that "For the introduction of this system [the 'graphic method'], which greatly simplifies difficult cases, I am indebted to Mr. Punnett." […] The first published diagrams appeared in 1906. […] when Punnett published the second edition of hisMendelism, he used a slightly different format ([…] Punnett 1907, p. 45) […] In the third edition (Punnett 1911, p. 34) he reverted to the arrangement […] with a description of the construction of what he called the "chessboard" method (although in truth it is more like a multiplication table). […] (11 pages)
^Gao, J.Z.; Fong, D.; Liu, X. (April 2011). "Mathematical analyses of casino rebate systems for VIP gambling".International Gambling Studies.11 (1):93–106.doi:10.1080/14459795.2011.552575.S2CID144540412.
^Gorman, Michael F. (2010). "Management Insights".Management Science.56:iv–vii.doi:10.1287/mnsc.1090.1132.
^Ross, Sheldon M. (2010).A First course in Probability (8th ed.). Pearson Prentice Hall. pp. 26–27.ISBN9780136033134.
^abWeisstein, Eric W."Probability".mathworld.wolfram.com. Retrieved10 September 2020.
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