How To Find Probability Between Two Numbers

Branch of mathematics concerning hazard and uncertainty

The probabilities of rolling several numbers using two dice.

Probability is the branch of mathematics apropos numerical descriptions of how likely an event is to occur, or how probable it is that a proffer is true. The probability of an issue is a number betwixt 0 and one, where, roughly speaking, 0 indicates impossibility of the issue and 1 indicates certainty.^{[annotation 1] [i] [2]} The college the probability of an event, the more likely it is that the event volition occur. A simple instance is the tossing of a fair (unbiased) coin. Since the coin is fair, the ii outcomes ("heads" and "tails") are both equally likely; 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/two (which could also exist written as 0.v or 50%).

These concepts have been given an evident mathematical formalization in probability theory, which is used widely in areas of study such as statistics, mathematics, scientific discipline, finance, gambling, artificial intelligence, machine learning, computer science, game theory, and philosophy to, for example, depict inferences nigh the expected frequency of events. Probability theory is likewise used to describe the underlying mechanics and regularities of circuitous systems.^[3]

Interpretations [edit]

When dealing with experiments that are random and well-defined in a purely theoretical setting (like tossing a coin), probabilities tin exist numerically described past the number of desired outcomes, divided by the full number of all outcomes. For instance, tossing a coin twice will yield "head-head", "head-tail", "tail-head", and "tail-tail" outcomes. The probability of getting an issue of "head-head" is i out of four outcomes, or, in numerical terms, 1/4, 0.25 or 25%. However, when it comes to practical application, there are two major competing categories of probability interpretations, whose adherents agree different views nearly the fundamental nature of probability:

• Objectivists assign numbers to describe some objective or physical country of affairs. The almost 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 the experiment is repeated indefinitely. This estimation considers probability to be the relative frequency "in the long run" of outcomes.^[4] A modification of this is propensity probability, which interprets probability as the tendency of some experiment to yield a certain effect, fifty-fifty if it is performed only once.
• Subjectivists assign numbers per subjective probability, that is, as a degree of belief.^[5] The caste of conventionalities has been interpreted equally "the cost at which you would buy or sell a bet that pays 1 unit of utility if Due east, 0 if non East",^[6] although that interpretation is not universally agreed upon.^[seven] The almost pop version of subjective probability is Bayesian probability, which includes proficient cognition too as experimental information to produce probabilities. The proficient noesis is represented by some (subjective) prior probability distribution. These data are incorporated in a likelihood part. The product of the prior and the likelihood, when normalized, results in a posterior probability distribution that incorporates all the information known to engagement.^[8] By Aumann's agreement theorem, Bayesian agents whose prior behavior are like will finish up with similar posterior beliefs. However, sufficiently unlike priors can lead to dissimilar conclusions, regardless of how much data the agents share.^[9]

Etymology [edit]

The give-and-take probability derives from the Latin probabilitas, which can likewise mean "probity", a measure of the potency of a witness in a legal case in Europe, and often correlated with the witness's dignity. 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.^[10]

History [edit]

The scientific study of probability is a mod development of mathematics. Gambling shows that there has been an interest in quantifying the ideas of probability for millennia, merely verbal mathematical descriptions arose much after. There are reasons for the ho-hum development of the mathematics of probability. Whereas games of risk provided the impetus for the mathematical study of probability, cardinal problems ^{[note ii]} are still obscured by the superstitions of gamblers.^[11]

Co-ordinate to Richard Jeffrey, "Before the eye of the seventeenth century, the term 'likely' (Latin probabilis) meant approvable, and was applied in that sense, univocally, to opinion and to action. A probable action or stance was one such as sensible people would undertake or concur, in the circumstances."^[12] However, in legal contexts specially, 'probable' could also apply to propositions for which there was adept evidence.^[thirteen]

The sixteenth-century Italian polymath Gerolamo Cardano demonstrated the efficacy of defining odds 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^[fourteen]). Aside from the elementary work by Cardano, 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.^[15] Jakob Bernoulli's Ars Conjectandi (posthumous, 1713) and Abraham de Moivre's Doctrine of Chances (1718) treated the discipline as a branch of mathematics.^[xvi] See Ian Hacking's The Emergence of Probability ^[10] and James Franklin's The Science of Conjecture ^[17] for histories of the early development of the very concept of mathematical probability.

The theory of errors may exist traced back to Roger Cotes's Opera Miscellanea (posthumous, 1722), but a memoir prepared by Thomas Simpson in 1755 (printed 1756) showtime applied the theory to the discussion of errors of observation.^[xviii] The reprint (1757) of this memoir lays downwardly the axioms that positive and negative errors are as probable, and that certain assignable limits define the range of all errors. Simpson also discusses continuous errors and describes a probability curve.

The outset ii laws of error that were proposed both originated with Pierre-Simon Laplace. The first police was published in 1774, and stated that the frequency of an mistake could be expressed as an exponential function of the numerical magnitude of the mistake—disregarding sign. The second law of error was proposed in 1778 by Laplace, and stated that the frequency of the fault is an exponential function of the square of the error.^[xix] The 2d police force of error is called the normal distribution or the Gauss law. "It is difficult historically to aspect that law to Gauss, who in spite of his well-known precocity had probably not made this discovery before he was 2 years one-time."^[nineteen]

Daniel Bernoulli (1778) introduced the principle of the maximum product of the probabilities of a organisation of concurrent errors.

Adrien-Marie Legendre (1805) developed the method of least squares, and introduced information technology in his Nouvelles méthodes pour la détermination des orbites des comètes (New Methods for Determining the Orbits of Comets).^[twenty] In ignorance of Legendre's contribution, an Irish-American author, Robert Adrain, editor of "The Analyst" (1808), first deduced the constabulary of facility of error,

${\displaystyle \phi (ten)=ce^{-h^{ii}x^{2}},}$

where ${\displaystyle h}$ is a constant depending on precision of observation, and ${\displaystyle c}$ is a scale factor ensuring that the area nether the bend equals 1. He gave 2 proofs, the 2nd being essentially the same equally John Herschel's (1850).^{[ citation needed ]} Gauss gave the first proof that seems to take been known in Europe (the third after Adrain'south) 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^{[ clarification needed ]} for r, the probable error of a unmarried 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.

In 1906, Andrey Markov introduced^[21] the notion of Markov chains, which played an important role in stochastic processes theory and its applications. The modern theory of probability based on the measure theory was developed by Andrey Kolmogorov in 1931.^[22]

On the geometric side, contributors to The Educational Times were influential (Miller, Crofton, McColl, Wolstenholme, Watson, and Artemas Martin).^[23] See integral geometry for more than info.

Theory [edit]

Like other theories, the theory of probability is a representation of its concepts in formal terms—that is, in terms that tin can exist considered separately from their meaning. These formal terms are manipulated by the rules of mathematics and logic, and whatsoever results are 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 (run into as well probability space), sets are interpreted as events and probability as a measure out on a form of sets. In Cox'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, the laws of probability are the aforementioned, except for technical details.

There are other methods for quantifying uncertainty, such every bit the Dempster–Shafer theory or possibility theory, simply those are substantially dissimilar and non uniform with the unremarkably-understood laws of probability.

Applications [edit]

Probability theory is applied in everyday life in risk assessment and modeling. The insurance industry and markets employ actuarial science to make up one's mind pricing and make trading decisions. Governments utilize probabilistic methods in environmental regulation, entitlement analysis, and financial regulation.

An case of the employ of probability theory in equity trading is the effect of the perceived probability of any widespread Centre East disharmonize on oil prices, which accept ripple effects in the economic system as a whole. An assessment by a commodity trader that a war is more likely can transport that commodity'south prices up or downwardly, and signals other traders of that stance. Appropriately, the probabilities are neither assessed independently nor necessarily rationally. The theory of behavioral finance emerged to draw the effect of such groupthink on pricing, on policy, and on peace and conflict.^[24]

In addition to financial assessment, probability can be used to clarify trends in biology (e.g., disease spread) as well every bit ecology (e.1000., biological Punnett squares). As with finance, risk assessment can exist used as a statistical tool to calculate the likelihood of undesirable events occurring, and can aid with implementing protocols to avoid encountering such circumstances. Probability is used to design games of chance then that casinos can make a guaranteed profit, yet provide payouts to players that are frequent enough to encourage connected play.^[25]

Another significant awarding of probability theory in everyday life is 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'southward decisions on a product's warranty.^[26]

The cache language model and other statistical linguistic communication models that are used in tongue processing are also examples of applications of probability theory.

Mathematical treatment [edit]

Calculation of probability (risk) vs odds

Consider an experiment that can produce a number of results. The drove of all possible results is chosen the sample space of the experiment, sometimes denoted as ${\displaystyle \Omega }$ . The power gear up of the sample space is formed by because all different collections of possible results. For example, rolling a die can produce 6 possible results. I collection of possible results gives an odd number on the dice. Thus, the subset {1,iii,5} is an chemical element of the power set up of the sample infinite of dice rolls. These collections are called "events". In this instance, {1,3,5} is the result that the dice falls on some odd number. If the results that really occur autumn in a given event, the consequence is said to have occurred.

A probability is a way of assigning every event a value betwixt nothing and i, with the requirement that the result made upwardly of all possible results (in our case, the upshot {1,2,3,4,5,6}) is assigned a value of one. To authorize as a probability, the assignment of values must satisfy the requirement that for whatever collection of mutually exclusive events (events with no common results, such every bit the events {1,6}, {three}, and {2,four}), the probability that at least one of the events volition occur is given by the sum of the probabilities of all the individual events.^[27]

The probability of an outcome A is written every bit ${\displaystyle P(A)}$ ,^[28] ${\displaystyle p(A)}$ , or ${\displaystyle {\text{Pr}}(A)}$ .^[29] This mathematical definition of probability can extend to infinite sample spaces, and even uncountable sample spaces, using the concept of a measure out.

The opposite or complement of an event A is the event [non A] (that is, the consequence of A not occurring), frequently denoted as ${\displaystyle A',A^{c}}$ , ${\displaystyle {\overline {A}},A^{\complement },\neg A}$ , or ${\displaystyle {\sim }A}$ ; its probability is given by P(not A) = ane − P(A).^[30] Every bit an example, the run a risk of not rolling a six on a six-sided die is i – (chance of rolling a half-dozen) ${\displaystyle =1-{\tfrac {one}{half dozen}}={\tfrac {5}{6}}}$ . For a more comprehensive handling, see Complementary event.

If two events A and B occur on a unmarried performance of an experiment, this is called the intersection or articulation probability of A and B, denoted equally ${\displaystyle P(A\cap B)}$ .

Independent events [edit]

If two events, A and B are independent so the articulation probability is^[28]

${\displaystyle P(A{\mbox{ and }}B)=P(A\cap B)=P(A)P(B).}$

For example, if two coins are flipped, and so the adventure of both existence heads is ${\displaystyle {\tfrac {1}{2}}\times {\tfrac {1}{2}}={\tfrac {1}{4}}}$ .^[31]

Mutually exclusive events [edit]

If either event A or event B can occur but never both simultaneously, and then they are called mutually exclusive events.

If two events are mutually exclusive, and so the probability of both occurring is denoted as ${\displaystyle P(A\cap B)}$ and

${\displaystyle P(A{\mbox{ and }}B)=P(A\cap B)=0}$

If two events are mutually exclusive, then the probability of either occurring is denoted every bit ${\displaystyle P(A\loving cup B)}$ and

${\displaystyle P(A{\mbox{ or }}B)=P(A\loving cup B)=P(A)+P(B)-P(A\cap B)=P(A)+P(B)-0=P(A)+P(B)}$

For example, the run a risk of rolling a 1 or 2 on a half dozen-sided die is ${\displaystyle P(one{\mbox{ or }}2)=P(ane)+P(2)={\tfrac {1}{half dozen}}+{\tfrac {i}{6}}={\tfrac {1}{3}}.}$

Not mutually sectional events [edit]

If the events are non mutually exclusive then

${\displaystyle P\left(A{\hbox{ or }}B\correct)=P(A\loving cup B)=P\left(A\right)+P\left(B\right)-P\left(A{\mbox{ and }}B\right).}$

For example, when cartoon a carte from a deck of cards, the adventure of getting a heart or a face card (J,Q,K) (or both) is ${\displaystyle {\tfrac {13}{52}}+{\tfrac {12}{52}}-{\tfrac {3}{52}}={\tfrac {11}{26}}}$ , since among the 52 cards of a deck, 13 are hearts, 12 are face cards, and iii are both: hither the possibilities included in the "3 that are both" are included in each of the "13 hearts" and the "12 face cards", simply should merely be counted once.

Conditional probability [edit]

Provisional probability is the probability of some event A, given the occurrence of another effect B. Conditional probability is written ${\displaystyle P(A\mid B)}$ , and is read "the probability of A, given B". It is defined by^[32]

${\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}.\,}$

If ${\displaystyle P(B)=0}$ then ${\displaystyle P(A\mid B)}$ is formally undefined by this expression. In this case ${\displaystyle A}$ and ${\displaystyle B}$ are independent, since ${\displaystyle P(A\cap B)=P(A)P(B)=0}$ . Still, it is possible to define a conditional probability for some nothing-probability events using a σ-algebra of such events (such as those arising from a continuous random variable).^{[ commendation needed ]}

For example, in a bag of two ruby-red balls and 2 bluish balls (4 balls in full), the probability of taking a ruby-red ball is ${\displaystyle 1/ii}$ ; nevertheless, when taking a second brawl, the probability of it existence either a scarlet ball or a blueish ball depends on the ball previously taken. For instance, if a cherry-red brawl was taken, then the probability of picking a ruby-red brawl again would be ${\displaystyle 1/3}$ , since just 1 ruby-red and 2 blue balls would accept been remaining. And if a blue ball was taken previously, the probability of taking a cherry-red ball will be ${\displaystyle 2/iii}$ .

Inverse probability [edit]

In probability theory and applications, Bayes' rule relates the odds of outcome ${\displaystyle A_{1}}$ to upshot ${\displaystyle A_{2}}$ , before (prior to) and after (posterior to) conditioning on another result ${\displaystyle B}$ . The odds on ${\displaystyle A_{1}}$ to event ${\displaystyle A_{two}}$ is simply the ratio of the probabilities of the two events. When arbitrarily many events ${\displaystyle A}$ are of involvement, not just two, the rule tin be rephrased as posterior is proportional to prior times likelihood, ${\displaystyle P(A|B)\propto P(A)P(B|A)}$ where the proportionality symbol ways that the left hand side is proportional to (i.e., equals a constant times) the right mitt side as ${\displaystyle A}$ varies, for stock-still or given ${\displaystyle B}$ (Lee, 2012; Bertsch McGrayne, 2012). In this form it goes dorsum to Laplace (1774) and to Cournot (1843); meet Fienberg (2005). Run into Changed probability and Bayes' rule.

Summary of probabilities [edit]

Summary of probabilities

Outcome	Probability
A	${\displaystyle P(A)\in [0,1]\,}$
not A	${\displaystyle P(A^{\complement })=one-P(A)\,}$
A or B	${\displaystyle {\brainstorm{aligned}P(A\cup B)&=P(A)+P(B)-P(A\cap B)\\P(A\cup B)&=P(A)+P(B)\qquad {\mbox{if A and B are mutually sectional}}\\\terminate{aligned}}}$
A and B	${\displaystyle {\begin{aligned}P(A\cap B)&=P(A|B)P(B)=P(B|A)P(A)\\P(A\cap B)&=P(A)P(B)\qquad {\mbox{if A and B are independent}}\\\end{aligned}}}$
A given B	${\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}={\frac {P(B|A)P(A)}{P(B)}}\,}$

Relation to randomness and probability in breakthrough mechanics [edit]

In a deterministic universe, based on Newtonian concepts, there would be no probability if all weather were known (Laplace's demon), (just in that location are situations in which sensitivity to initial conditions exceeds our ability to measure them, i.e. know them). In the instance of a roulette wheel, if the force of the hand and the period of that strength are known, the number on which the ball volition stop would be a certainty (though as a applied matter, this would probable be true but of a roulette bicycle that had not been exactly levelled – every bit Thomas A. Bass' Newtonian Casino revealed). This likewise assumes noesis of inertia and friction of the wheel, weight, smoothness, and roundness of the ball, variations in paw speed during the turning, and then 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 confront the same situation in the kinetic theory of gases, where the organization, while deterministic in principle, is so complex (with the number of molecules typically the guild of magnitude of the Avogadro abiding vi.02×10²³ ) that only a statistical description of its backdrop is feasible.

Probability theory is required to describe quantum phenomena.^[33] A revolutionary discovery of early 20th century physics was the random grapheme of all physical processes that occur at sub-diminutive scales and are governed past the laws of quantum mechanics. The objective wave function evolves deterministically but, co-ordinate to the Copenhagen interpretation, it deals with probabilities of observing, the upshot existence explained by a wave role plummet when an observation is made. However, the loss of determinism for the sake of instrumentalism did not run into with universal approval. Albert Einstein famously remarked in a letter to Max Built-in: "I am convinced that God does non play dice".^[34] Like Einstein, Erwin Schrödinger, who discovered the wave function, believed quantum mechanics is a statistical approximation of an underlying deterministic reality.^[35] In some mod interpretations of the statistical mechanics of measurement, quantum decoherence is invoked to account for the appearance of subjectively probabilistic experimental outcomes.

See also [edit]

• Take chances (disambiguation)
• Class membership probabilities
• Contingency
• Equiprobability
• Heuristics in judgment and controlling
• Probability theory
• Randomness
• Statistics
• Estimators
• Estimation theory
• Probability density part
• Pairwise independence

In law

• Residual of probabilities

Notes [edit]

1. ^ Strictly speaking, a probability of 0 indicates that an issue about never takes identify, whereas a probability of 1 indicates than an consequence well-nigh certainly takes place. This is an important stardom when the sample space is infinite. For example, for the continuous compatible distribution on the real interval [5, 10], there are an infinite number of possible outcomes, and the probability of whatever given issue being observed — for example, exactly 7 — is 0. This means that when we make an observation, it volition almost surely not be exactly 7. However, it does not mean that exactly 7 is impossible. Ultimately some specific result (with probability 0) will be observed, and one possibility for that specific outcome is exactly 7.
2. ^ 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.

References [edit]

1. ^ "Kendall's Avant-garde Theory of Statistics, Volume ane: Distribution Theory", Alan Stuart and Keith Ord, sixth Ed, (2009), ISBN 978-0-534-24312-8.
2. ^ William Feller, An Introduction to Probability Theory and Its Applications, (Vol 1), third Ed, (1968), Wiley, ISBN 0-471-25708-seven.
3. ^ Probability Theory The Britannica website
4. ^ Hacking, Ian (1965). The Logic of Statistical Inference. Cambridge Academy Press. ISBN978-0-521-05165-1. ^{[ page needed ]}
5. ^ Finetti, Bruno de (1970). "Logical foundations and measurement of subjective probability". Acta Psychologica. 34: 129–145. doi:ten.1016/0001-6918(seventy)90012-0.
6. ^ Hájek, Alan (21 October 2002). Edward N. Zalta (ed.). "Interpretations of Probability". The Stanford Encyclopedia of Philosophy (Winter 2012 ed.). Retrieved 22 Apr 2013.
7. ^ Jaynes, Eastward.T. (2003). "Department A.2 The de Finetti system of probability". In Bretthorst, G. Larry (ed.). Probability Theory: The Logic of Science (i ed.). Cambridge University Press. ISBN978-0-521-59271-0.
8. ^ Hogg, Robert Five.; Craig, Allen; McKean, Joseph Westward. (2004). Introduction to Mathematical Statistics (6th ed.). Upper Saddle River: Pearson. ISBN978-0-xiii-008507-viii. ^{[ page needed ]}
9. ^ Jaynes, E.T. (2003). "Section 5.3 Converging and diverging views". In Bretthorst, G. Larry (ed.). Probability Theory: The Logic of Scientific discipline (ane ed.). Cambridge University Press. ISBN978-0-521-59271-0.
10. ^ ^{a b} 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^{[ page needed ]}
11. ^ Freund, John. (1973) Introduction to Probability. Dickenson ISBN 978-0-8221-0078-two (p. i)
12. ^ Jeffrey, R.C., Probability and the Art of Judgment, Cambridge University Press. (1992). pp. 54–55 . ISBN 0-521-39459-seven
13. ^ Franklin, J. (2001) The Science of Theorize: Show and Probability Before Pascal, Johns Hopkins University Press. (pp. 22, 113, 127)
14. ^ "Some laws and problems in classical probability and how Cardano anticipated them Gorrochum, P. Risk magazine 2012" (PDF).
15. ^ Abrams, William, A Brief History of Probability, 2d Moment, retrieved 23 May 2008
16. ^ Ivancevic, Vladimir G.; Ivancevic, Tijana T. (2008). Quantum leap : from Dirac and Feynman, across the universe, to human torso and mind. Singapore ; Hackensack, NJ: World Scientific. p. 16. ISBN978-981-281-927-7.
17. ^ Franklin, James (2001). The Science of Theorize: Evidence and Probability Before Pascal. Johns Hopkins University Press. ISBN978-0-8018-6569-5.
18. ^ Shoesmith, Eddie (Nov 1985). "Thomas Simpson and the arithmetic mean". Historia Mathematica. 12 (4): 352–355. doi:10.1016/0315-0860(85)90044-8.
19. ^ ^{a b} Wilson EB (1923) "First and second laws of error". Journal of the American Statistical Clan, 18, 143
20. ^ Seneta, Eugene William. ""Adrien-Marie Legendre" (version 9)". StatProb: The Encyclopedia Sponsored by Statistics and Probability Societies. Archived from the original on 3 February 2016. Retrieved 27 January 2016.
21. ^ Weber, Richard. "Markov Bondage" (PDF). Statistical Laboratory. University of Cambridge.
22. ^ Vitanyi, Paul M.B. (1988). "Andrei Nikolaevich Kolmogorov". CWI Quarterly (ane): three–18. Retrieved 27 January 2016.
23. ^ Wilcox, Rand R. (ten May 2016). Understanding and applying basic statistical methods using R. Hoboken, New Jersey. ISBN978-1-119-06140-iii. OCLC 949759319.
24. ^ Singh, Laurie (2010) "Whither Efficient Markets? Efficient Market Theory and Behavioral Finance". The Finance Professionals' Post, 2010.
25. ^ Gao, J.Z.; Fong, D.; Liu, X. (Apr 2011). "Mathematical analyses of casino rebate systems for VIP gambling". International Gambling Studies. 11 (1): 93–106. doi:ten.1080/14459795.2011.552575. S2CID 144540412.
26. ^ Gorman, Michael F. (2010). "Direction Insights". Direction Science. 56: 4–vii. doi:10.1287/mnsc.1090.1132.
27. ^ Ross, Sheldon Grand. (2010). A Starting time form in Probability (8th ed.). Pearson Prentice Hall. pp. 26–27. ISBN9780136033134.
28. ^ ^{a b} Weisstein, Eric W. "Probability". mathworld.wolfram.com . Retrieved x September 2020.
29. ^ Olofsson (2005) p. viii.
30. ^ Olofsson (2005), p. 9
31. ^ Olofsson (2005) p. 35.
32. ^ Olofsson (2005) p. 29.
33. ^ Burgin, Mark (2010). "Interpretations of Negative Probabilities". p. 1. arXiv:1008.1287v1 [physics.data-an].
34. ^ Jedenfalls bin ich überzeugt, daß der Alte nicht würfelt. Letter to Max Built-in, four December 1926, in: Einstein/Built-in Briefwechsel 1916–1955.
35. ^ Moore, W.J. (1992). Schrödinger: Life and Thought. Cambridge University Press. p. 479. ISBN978-0-521-43767-7.

Bibliography [edit]

• Kallenberg, O. (2005) Probabilistic Symmetries and Invariance Principles. Springer-Verlag, New York. 510 pp. ISBN 0-387-25115-four
• Kallenberg, O. (2002) Foundations of Modernistic Probability, 2nd ed. Springer Series in Statistics. 650 pp. ISBN 0-387-95313-2
• Olofsson, Peter (2005) Probability, Statistics, and Stochastic Processes, Wiley-Interscience. 504 pp ISBN 0-471-67969-0.

External links [edit]

• Virtual Laboratories in Probability and Statistics (Univ. of Ala.-Huntsville)
• Probability on In Our Fourth dimension at the BBC
• Probability and Statistics EBook
• Edwin Thompson Jaynes. Probability Theory: The Logic of Science. Preprint: Washington University, (1996). — HTML index with links to PostScript files and PDF (first three chapters)
• People from the History of Probability and Statistics (Univ. of Southampton)
• Probability and Statistics on the Earliest Uses Pages (Univ. of Southampton)
• Primeval Uses of Symbols in Probability and Statistics on Earliest Uses of Various Mathematical Symbols
• A tutorial on probability and Bayes' theorem devised for first-twelvemonth Oxford Academy students
• [1] pdf file of An Album of Chance Operations (1963) at UbuWeb
• Introduction to Probability – eBook, by Charles Grinstead, Laurie Snell Source Archived 25 March 2012 at the Wayback Automobile (GNU Free Documentation License)
• (in English language and Italian) Bruno de Finetti, Probabilità e induzione, Bologna, CLUEB, 1993. ISBN 88-8091-176-7 (digital version)
• Richard P. Feynman's Lecture on probability.

Source: https://en.wikipedia.org/wiki/Probability

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