Misplaced Pages

#819180

87-415: HEART method is based upon the principle that every time a task is performed there is a possibility of failure and that the probability of this is affected by one or more Error Producing Conditions (EPCs) – for instance: distraction, tiredness, cramped conditions etc. – to varying degrees. Factors which have a significant effect on performance are of greatest interest. These conditions can then be applied to

174-594: A casino setting. Gambling games that take place outside of casinos include bingo (as played in the US and UK ), dead pool , lotteries , pull-tab games and scratchcards , and Mahjong . Other non-casino gambling games include: *Although coin tossing is not usually played in a casino, it has been known to be an official gambling game in some Australian casinos Fixed-odds betting and Parimutuel betting frequently occur at many types of sporting events, and political elections. In addition many bookmakers offer fixed odds on

261-441: A random event with the intent of winning something else of value, where instances of strategy are discounted. Gambling thus requires three elements to be present: consideration (an amount wagered), risk (chance), and a prize . The outcome of the wager is often immediate, such as a single roll of dice , a spin of a roulette wheel, or a horse crossing the finish line, but longer time frames are also common, allowing wagers on

348-425: A σ-algebra of such events (such as those arising from a continuous random variable ). 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 1 / 2 ; {\displaystyle 1/2;} 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

435-537: A "best-case-scenario" estimate of the failure probability under ideal conditions to then obtain a final error chance. This figure assists in communication of error chances with the wider risk analysis or safety case. By forcing consideration of the EPCs potentially affecting a given procedure, HEART also has the indirect effect of providing a range of suggestions as to how the reliability may therefore be improved (from an ergonomic standpoint) and hence minimising risk. HEART

522-421: A card from a deck of cards, the chance of getting a heart or a face card (J, Q, K) (or both) is 13 52 + 12 52 − 3 52 = 11 26 , {\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 3 are both: here

609-488: A close connection between many governments and gambling organizations, where legal gambling provides significant government revenue, such as in Monaco and Macau , China. There is generally legislation requiring that gambling devices be statistically random , to prevent manufacturers from making some high-payoff results impossible. Since these high payoffs have very low probability , a house bias can quite easily be missed unless

696-407: A favorite theme for over three centuries. It has been heavily regulated. Historically much of the opposition comes from Nonconformist Protestants , and from social reformers. Gambling has been part of Singapore's history, though it was strictly controlled by the government for many years. In the mid-20th century, illegal gambling was common. However, with the opening of regulated casinos in 2010,

783-648: A gambling contract may not give a casino bona fide purchaser status, permitting the recovery of stolen funds in some situations. In Lipkin Gorman v Karpnale Ltd , where a solicitor used stolen funds to gamble at a casino, the House of Lords overruled the High Court's previous verdict, adjudicating that the casino return the stolen funds less those subject to any change of position defence. U.S. Law precedents are somewhat similar. For case law on recovery of gambling losses where

870-478: A manufacturer's decisions on a product's warranty . The cache language model and other statistical language models that are used in natural language processing are also examples of applications of probability theory. Consider an experiment that can produce a number of results. The collection of all possible results is called the sample space of the experiment, sometimes denoted as Ω {\displaystyle \Omega } . The power set of

957-587: A number of non-sports related outcomes, for example the direction and extent of movement of various financial indices , the winner of television competitions such as Big Brother , and election results. Interactive prediction markets also offer trading on these outcomes, with "shares" of results trading on an open market. One of the most widespread forms of gambling involves betting on horse or greyhound racing . Wagering may take place through parimutuel pools, or bookmakers may take bets personally. Parimutuel wagers pay off at prices determined by support in

SECTION 10

#1732859488820

1044-426: A red ball was taken, then the probability of picking a red ball again would be 1 / 3 , {\displaystyle 1/3,} 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 2 / 3. {\displaystyle 2/3.} In probability theory and applications, Bayes' rule relates

1131-750: A source of destruction in Singalovada Sutra . Professions that are seen to violate the precept against theft include working in the gambling industry. Ancient Hindu poems like the Gambler's Lament and the Mahabharata testify to the existence of gambling among ancient Indians, while highlighting its destructive impact. The text Arthashastra ( c.  4th century BCE ) recommends taxation and control of gambling. Ancient Jewish authorities frowned on gambling, even disqualifying professional gamblers from testifying in court. The Catholic Church holds

1218-430: A statement is true or false, or that a specified event will happen (a "back bet") or will not happen (a "lay bet") within a specified time. This occurs in particular when two people have opposing but strongly held views on truth or events. Not only do the parties hope to gain from the bet, they place the bet also to demonstrate their certainty about the issue. Some means of determining the issue at stake must exist. Sometimes

1305-492: A value, but are not real money. For example, players of marbles games might wager marbles, and likewise games of Pogs or Magic: The Gathering can be played with the collectible game pieces (respectively, small discs and trading cards) as stakes, resulting in a meta-game regarding the value of a player's collection of pieces. Gambling dates back at least to the Paleolithic period, before written history. In Mesopotamia

1392-585: Is a consensus among the ‘ Ulema ’ ( Arabic : عُـلـمـاء , Scholars (of Islam )) that gambling is haraam ( Arabic : حَـرام , sinful or forbidden). In assertions made during its prohibition, Muslim jurists describe gambling as being both un- Qur’anic , and as being generally harmful to the Muslim Ummah ( Arabic : أُمَّـة , Community). The Arabic terminology for gambling is Maisir . They ask you about intoxicants and gambling. Say: 'In them both lies grave sin, though some benefit, to mankind. But their sin

1479-464: Is a theoretically risk-free betting system in which every outcome of an event is bet upon so that a known profit will be made by the bettor upon completion of the event regardless of the outcome. Arbitrage betting is a combination of the ancient art of arbitrage trading and gambling, which has been made possible by the large numbers of bookmakers in the marketplace, creating occasional opportunities for arbitrage. One can also bet with another person that

1566-543: Is acceptable is a matter of debate: Investments are also usually not considered gambling, although some investments can involve significant risk. Examples of investments include stocks , bonds and real estate . Starting a business can also be considered a form of investment. Investments are generally not considered gambling when they meet the following criteria: Some speculative investment activities are particularly risky, but are sometimes perceived to be different from gambling: A levant or levanting characterises

1653-422: Is arrived at from inductive reasoning and statistical inference . 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

1740-544: Is denoted as P ( A ∪ B ) {\displaystyle P(A\cup B)} and P ( A  or  B ) = P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) = P ( A ) + P ( B ) − 0 = P ( A ) + P ( B ) {\displaystyle P(A{\mbox{ or }}B)=P(A\cup B)=P(A)+P(B)-P(A\cap B)=P(A)+P(B)-0=P(A)+P(B)} For example,

1827-401: Is established. 3. The EPCs, which are apparent in the given situation and highly probable to have a negative effect on the outcome, are then considered and the extent to which each EPC applies to the task in question is discussed and agreed, again with local experts. As an EPC should never be considered beneficial to a task, it is calculated using the following formula: 4. A final estimate of

SECTION 20

#1732859488820

1914-446: Is formally undefined by this expression. In this case A {\displaystyle A} and B {\displaystyle B} are independent, since P ( A ∩ B ) = P ( A ) P ( B ) = 0. {\displaystyle P(A\cap B)=P(A)P(B)=0.} However, it is possible to define a conditional probability for some zero-probability events, for example by using

2001-579: Is given by P (not A ) = 1 − P ( A ) . As an example, the chance of not rolling a six on a six-sided die is 1 – (chance of rolling a six) = 1 − ⁠ 1 / 6 ⁠ = ⁠ 5 / 6 ⁠ . For a more comprehensive treatment, see Complementary event . If two events A and B occur on a single performance of an experiment, this is called the intersection or joint probability of A and B , denoted as P ( A ∩ B ) . {\displaystyle P(A\cap B).} If two events, A and B are independent then

2088-616: Is more grave than their benefit.' In parts of the world that implement full Shari‘ah, such as Aceh , punishments for Muslim gamblers can range up to 12 lashes or a one-year prison term and a fine for those who provide a venue for such practises. Some Islamic nations prohibit gambling; most other countries regulate it . According to the Most Holy Book , paragraph 155, gambling is forbidden. While almost any game can be played for money, and any game typically played for money can also be played just for fun, some games are generally offered in

2175-663: Is not universally observed in the English-speaking world. For instance, in the United Kingdom, the regulator of gambling activities is called the Gambling Commission (not the Gaming Commission). The word gaming is used more frequently since the rise of computer and video games to describe activities that do not necessarily involve wagering, especially online gaming , with the new usage still not having displaced

2262-508: Is reached, and the motivation is entertainment and not personal gain leading to the "love of money" or making a living. In general, Catholic bishops have opposed casino gambling on the grounds that it too often tempts people into problem gambling or addiction, and has particularly negative effects on poor people; they sometimes also cite secondary effects such as increases in loan sharking, prostitution, corruption, and general public immorality. Some parish pastors have also opposed casinos for

2349-542: Is referred to as theoretical probability (in contrast to empirical probability , dealing with probabilities in the context of real experiments). 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%. However, when it comes to practical application, there are two major competing categories of probability interpretations, whose adherents hold different views about

2436-420: Is simply the ratio of the probabilities of the two events. When arbitrarily many events A {\displaystyle A} are of interest, not just two, the rule can be rephrased as posterior is proportional to prior times likelihood , P ( A | B ) ∝ P ( A ) P ( B | A ) {\displaystyle P(A|B)\propto P(A)P(B|A)} where

2523-640: Is the probability of some event A , given the occurrence of some other event B . Conditional probability is written P ( A ∣ B ) {\displaystyle P(A\mid B)} , and is read "the probability of A , given B ". It is defined by P ( A ∣ B ) = P ( A ∩ B ) P ( B ) {\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}\,} If P ( B ) = 0 {\displaystyle P(B)=0} then P ( A ∣ B ) {\displaystyle P(A\mid B)}

2610-443: Is used to design games of chance so that casinos can make a guaranteed profit, yet provide payouts to players that are frequent enough to encourage continued play. Another significant application 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

2697-981: The Christian Reformed Church in North America , the Church of the Lutheran Confession , the Southern Baptist Convention , the Assemblies of God , and the Seventh-day Adventist Church . Other churches that oppose gambling include the Jehovah's Witnesses , The Church of Jesus Christ of Latter-day Saints , the Iglesia ni Cristo , and the Members Church of God International . There

Human error assessment and reduction technique - Misplaced Pages Continue

2784-538: 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 remarked in a letter to Max Born : "I am convinced that God does not play dice". Like Einstein, Erwin Schrödinger , who discovered

2871-498: The Kolmogorov formulation and the Cox formulation. In Kolmogorov's formulation (see also probability space ), sets are interpreted as events and probability as a measure on a class 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,

2958-680: The laws of probability are the same, except for technical details. 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 usually-understood laws of probability. Probability theory is applied in everyday life in risk assessment and modeling . The insurance industry and markets use actuarial science to determine pricing and make trading decisions. Governments apply probabilistic methods in environmental regulation , entitlement analysis, and financial regulation . An example of

3045-426: The odds of event A 1 {\displaystyle A_{1}} to event A 2 , {\displaystyle A_{2},} before (prior to) and after (posterior to) conditioning on another event B . {\displaystyle B.} The odds on A 1 {\displaystyle A_{1}} to event A 2 {\displaystyle A_{2}}

3132-431: The probable error of a single observation, is well known. In the nineteenth century, authors on the general theory included Laplace , Sylvestre Lacroix (1816), Littrow (1833), Adolphe Quetelet (1853), Richard Dedekind (1860), Helmert (1872), Hermann Laurent (1873), Liagre, Didion and Karl Pearson . Augustus De Morgan and George Boole improved the exposition of the theory. In 1906, Andrey Markov introduced

3219-402: The theory 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

3306-607: The "bet-upon" outcome beyond the specific financial terms; for example, a "bet" with an insurer on whether one's house will burn down is not gambling, but rather insurance , as the homeowner has an obvious interest in the continued existence of the home independent of the purely financial aspects of the "bet" (i.e., the insurance policy). Nonetheless, both insurance and gambling contracts are typically considered aleatory contracts under most legal systems, though they are subject to different types of regulation. Under common law , particularly English Law ( English unjust enrichment ),

3393-620: The 14th century. Poker , the most popular U.S. card game associated with gambling, derives from the Persian game As-Nas , dating back to the 17th century. The first known casino, the Ridotto , started operating in 1638 in Venice, Italy. Gambling has been a main recreational activity in Great Britain for centuries. Queen Elizabeth I chartered a lottery that was drawn in 1569. Horseracing has been

3480-459: The HEP is then calculated, in determination of which the identified EPC's play a large part. Only those EPC's which show much evidence with regards to their affect in the contextual situation should be used by the assessor. A reliability engineer has the task of assessing the probability of a plant operator failing to carry out the task of isolating a plant bypass route as required by procedure. However,

3567-621: The act of absconding following the outcome of a bet. Problem gambling has multiple symptoms. Gamblers often play again to try to win back money they have lost, and some gamble to relieve feelings of helplessness and anxiety. In the United Kingdom, the Advertising Standards Authority has censured several betting firms for advertisements disguised as news articles suggesting falsely that a person had cleared debts and paid for medical expenses by gambling online. The firms face possible fines. A 2020 study of 32 countries found that

Human error assessment and reduction technique - Misplaced Pages Continue

3654-713: The additional reason that they would take customers away from church bingo and annual festivals where games such as blackjack , roulette , craps , and poker are used for fundraising. St. Thomas Aquinas wrote that gambling should be especially forbidden where the losing bettor is underage or otherwise not able to consent to the transaction. Gambling has often been seen as having social consequences , as satirized by Balzac . For these social and religious reasons, most legal jurisdictions limit gambling, as advocated by Pascal . Gambling views among Protestants vary, with some either discouraging or forbidding their members from participation in gambling. Methodists , in accordance with

3741-413: The amount bet remains nominal, demonstrating the outcome as one of principle rather than of financial importance. Betting exchanges allow consumers to both back and lay at odds of their choice. Similar in some ways to a stock exchange, a bettor may want to back a horse (hoping it will win) or lay a horse (hoping it will lose, effectively acting as bookmaker). Spread betting allows gamblers to wager on

3828-479: The approach shifted. Today, the government enforces strict laws to promote responsible gambling and prevent illegal activities. Gambling has been a popular activity in the United States for centuries. It has also been suppressed by law in many areas for almost as long. By the early 20th century, gambling was almost uniformly outlawed throughout the U.S. and thus became a largely illegal activity, helping to spur

3915-399: 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 the kinetic theory of gases , where the system, while deterministic in principle , is so complex (with the number of molecules typically

4002-428: 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' Newtonian Casino revealed). This also assumes knowledge of inertia and friction of the wheel, weight, smoothness, and roundness of

4089-1116: The chance of rolling a 1 or 2 on a six-sided die is P ( 1  or  2 ) = P ( 1 ) + P ( 2 ) = 1 6 + 1 6 = 1 3 . {\displaystyle P(1{\mbox{ or }}2)=P(1)+P(2)={\tfrac {1}{6}}+{\tfrac {1}{6}}={\tfrac {1}{3}}.} If the events are not (necessarily) mutually exclusive then P ( A  or  B ) = P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A  and  B ) . {\displaystyle P\left(A{\hbox{ or }}B\right)=P(A\cup B)=P\left(A\right)+P\left(B\right)-P\left(A{\mbox{ and }}B\right).} Rewritten, P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) {\displaystyle P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)} For example, when drawing

4176-446: The circumstances." However, in legal contexts especially, 'probable' could also apply to propositions for which there was good evidence. 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 ). Aside from

4263-636: 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%). These concepts have been given an axiomatic mathematical formalization in probability theory , which is used widely in areas of study such as statistics , mathematics , science , finance , gambling , artificial intelligence , machine learning , computer science , game theory , and philosophy to, for example, draw inferences about

4350-441: The collective product of the factors. 1. The first stage of the process is to identify the full range of sub-tasks that a system operator would be required to complete within a given task. 2. Once this task description has been constructed a nominal human unreliability score for the particular task is then determined, usually by consulting local experts. Based around this calculated point, a 5th – 95th percentile confidence range

4437-500: The concept of a measure. The opposite or complement of an event A is the event [not A ] (that is, the event of A not occurring), often denoted as A ′ , A c {\displaystyle A',A^{c}} , A ¯ , A ∁ , ¬ A {\displaystyle {\overline {A}},A^{\complement },\neg A} , or ∼ A {\displaystyle {\sim }A} ; its probability

SECTION 50

#1732859488820

4524-603: The curve equals 1. He gave two proofs, the second being essentially the same as John Herschel 's (1850). 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), and Morgan Crofton (1870). Other contributors were Ellis (1844), De Morgan (1864), Glaisher (1872), and Giovanni Schiaparelli (1875). Peters 's (1856) formula for r ,

4611-516: The devices are checked carefully. Most jurisdictions that allow gambling require participants to be above a certain age. In some jurisdictions, the gambling age differs depending on the type of gambling. For example, in many American states one must be over 21 to enter a casino, but may buy a lottery ticket after turning 18. Because contracts of insurance have many features in common with wagers, insurance contracts are often distinguished in law as agreements in which either party has an interest in

4698-708: The doctrine of outward holiness , oppose gambling which they believe is a sin that feeds on greed. Other denominations that discourage gambling are the United Methodist Church , the Free Methodist Church , the Evangelical Wesleyan Church , the Salvation Army , and the Church of the Nazarene . Other Protestants that oppose gambling include Mennonites , Schwarzenau Brethren , Quakers ,

4785-564: The earliest six-sided dice date to about 3000  BCE . However, they were based on astragali dating back thousands of years earlier. In China, gambling houses were widespread in the first millennium BCE, and betting on fighting animals was common. Lotto games and dominoes (precursors of Pai Gow ) appeared in China as early as the 10th century. Playing cards appeared in the 9th century CE in China. Records trace gambling in Japan back at least as far as

4872-461: The early development of the very concept of mathematical probability. The theory of errors may be traced back to Roger Cotes 's Opera Miscellanea (posthumous, 1722), but a memoir prepared by Thomas Simpson in 1755 (printed 1756) first applied the theory to the discussion of errors of observation. The reprint (1757) of this memoir lays down the axioms that positive and negative errors are equally probable, and that certain assignable limits define

4959-480: The effect of such groupthink on pricing, on policy, and on peace and conflict. In addition to financial assessment, probability can be used to analyze trends in biology (e.g., disease spread) as well as ecology (e.g., biological Punnett squares ). 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

5046-515: 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. Jakob Bernoulli 's Ars Conjectandi (posthumous, 1713) and Abraham de Moivre 's Doctrine of Chances (1718) treated the subject as a branch of mathematics. See Ian Hacking 's The Emergence of Probability and James Franklin's The Science of Conjecture for histories of

5133-554: 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. The probability of an event A is written as P ( A ) {\displaystyle P(A)} , p ( A ) {\displaystyle p(A)} , or Pr ( A ) {\displaystyle {\text{Pr}}(A)} . This mathematical definition of probability can extend to infinite sample spaces, and even uncountable sample spaces, using

5220-427: The expected frequency of events. Probability theory is also used to describe the underlying mechanics and regularities of complex systems . When dealing with random experiments – i.e., experiments that are random and well-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

5307-459: The fundamental nature of probability: The word probability derives from the Latin probabilitas , which can also mean " 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

SECTION 60

#1732859488820

5394-472: The growth of the mafia and other criminal organizations . The late 20th century saw a softening in attitudes towards gambling and a relaxation of laws against it. Many jurisdictions, local as well as national, either ban gambling or heavily control it by licensing the vendors. Such regulation generally leads to gambling tourism and illegal gambling in the areas where it is not allowed. The involvement of governments, through regulation and taxation, has led to

5481-413: The impetus for the mathematical study of probability, fundamental issues are still obscured by superstitions. 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

5568-690: The joint probability is P ( A  and  B ) = P ( A ∩ B ) = P ( A ) P ( B ) . {\displaystyle P(A{\mbox{ and }}B)=P(A\cap B)=P(A)P(B).} For example, if two coins are flipped, then the chance of both being heads is 1 2 × 1 2 = 1 4 . {\displaystyle {\tfrac {1}{2}}\times {\tfrac {1}{2}}={\tfrac {1}{4}}.} If either event A or event B can occur but never both simultaneously, then they are called mutually exclusive events. If two events are mutually exclusive , then

5655-400: The law of facility of error, ϕ ( x ) = c e − h 2 x 2 {\displaystyle \phi (x)=ce^{-h^{2}x^{2}}} where h {\displaystyle h} is a constant depending on precision of observation, and c {\displaystyle c} is a scale factor ensuring that the area under

5742-552: The loser had stolen the funds see "Rights of owner of stolen money as against one who won it in gambling transaction from thief". An interesting question is what happens when the person trying to make recovery is the gambler's spouse, and the money or property lost was either the spouse's, or was community property . This was a minor plot point in a Perry Mason novel, The Case of the Singing Skirt , and it cites an actual case Novo v. Hotel Del Rio . The Buddha stated gambling as

5829-408: The normal likelihood of failure can therefore be formulated as: Probability Probability is the branch of mathematics concerning events 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. A simple example is the tossing of a fair (unbiased) coin. Since

5916-437: The notion of Markov chains , which played an important role in stochastic processes theory and its applications. The modern theory of probability based on measure theory was developed by Andrey Kolmogorov in 1931. On the geometric side, contributors to The Educational Times included Miller, Crofton, McColl, Wolstenholme, Watson, and Artemas Martin . See integral geometry for more information. Like other theories ,

6003-509: The old usage as the primary definition in common dictionaries. "Gaming" has also been used euphemistically to circumvent laws against "gambling". The media and others have used one term or the other to frame conversations around the subjects, resulting in a shift of perceptions among their audiences. Gambling is also a major international commercial activity, with the legal gambling market totaling an estimated $ 335 billion in 2009. In other forms, gambling can be conducted with materials that have

6090-449: The operator is fairly inexperienced in fulfilling this task and therefore typically does not follow the correct procedure; the individual is therefore unaware of the hazards created when the task is carried out There are various assumptions that should be considered in the context of the situation: A representation of this situation using the HEART methodology would be done as follows: From

6177-516: The order of magnitude of the Avogadro constant 6.02 × 10 ) that only a statistical description of its properties is feasible. Probability theory is required to describe quantum phenomena. 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

6264-527: The outcome of a future sports contest or even an entire sports season. The term "gaming" in this context typically refers to instances in which the activity has been specifically permitted by law . The two words are not mutually exclusive; i.e. , a "gaming" company offers (legal) "gambling" activities to the public and may be regulated by one of many gaming control boards , for example, the Nevada Gaming Control Board . However, this distinction

6351-605: The outcome of an event where the pay-off is based on the accuracy of the wager, rather than a simple "win or lose" outcome. For example, a wager can be based on the when a point is scored in the game in minutes and each minute away from the prediction increases or reduces the payout. Many betting systems have been created in an attempt to "beat the house" but no system can make a mathematically unprofitable bet in terms of expected value profitable over time. Widely used systems include: Many risk-return choices are sometimes referred to colloquially as "gambling." Whether this terminology

6438-401: The position that there is no moral impediment to gambling, so long as it is fair, all bettors have a reasonable chance of winning, there is no fraud involved, and the parties involved do not have actual knowledge of the outcome of the bet (unless they have disclosed this knowledge), and as long as the following conditions are met: the gambler can afford to lose the bet, and stops when the limit

6525-2599: 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: P ( A ∪ B ∪ C ) = P ( ( A ∪ B ) ∪ C ) = P ( A ∪ B ) + P ( C ) − P ( ( A ∪ B ) ∩ C ) = P ( A ) + P ( B ) − P ( A ∩ B ) + P ( C ) − P ( ( A ∩ C ) ∪ ( B ∩ C ) ) = P ( A ) + P ( B ) + P ( C ) − P ( A ∩ B ) − ( P ( A ∩ C ) + P ( B ∩ C ) − P ( ( A ∩ C ) ∩ ( B ∩ C ) ) ) P ( A ∪ B ∪ C ) = P ( A ) + P ( B ) + P ( C ) − P ( A ∩ B ) − P ( A ∩ C ) − P ( B ∩ C ) + P ( A ∩ B ∩ C ) {\displaystyle {\begin{aligned}P\left(A\cup B\cup C\right)=&P\left(\left(A\cup B\right)\cup C\right)\\=&P\left(A\cup B\right)+P\left(C\right)-P\left(\left(A\cup B\right)\cap C\right)\\=&P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)+P\left(C\right)-P\left(\left(A\cap C\right)\cup \left(B\cap C\right)\right)\\=&P\left(A\right)+P\left(B\right)+P\left(C\right)-P\left(A\cap B\right)-\left(P\left(A\cap C\right)+P\left(B\cap C\right)-P\left(\left(A\cap C\right)\cap \left(B\cap C\right)\right)\right)\\P\left(A\cup B\cup C\right)=&P\left(A\right)+P\left(B\right)+P\left(C\right)-P\left(A\cap B\right)-P\left(A\cap C\right)-P\left(B\cap C\right)+P\left(A\cap B\cap C\right)\end{aligned}}} It can be seen, then, that this pattern can be repeated for any number of events. Conditional probability

6612-491: The principle of the maximum product of the probabilities of a system of concurrent errors. 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 ). In ignorance of Legendre's contribution, an Irish-American writer, Robert Adrain , editor of "The Analyst" (1808), first deduced

6699-399: The probability of both occurring is denoted as P ( A ∩ B ) {\displaystyle P(A\cap B)} and P ( A  and  B ) = P ( A ∩ B ) = 0 {\displaystyle P(A{\mbox{ and }}B)=P(A\cap B)=0} If two events are mutually exclusive , then the probability of either occurring

6786-646: The proportionality symbol means that the left hand side is proportional to (i.e., equals a constant times) the right hand side as A {\displaystyle A} varies, for fixed or given B {\displaystyle B} (Lee, 2012; Bertsch McGrayne, 2012). In this form it goes back to Laplace (1774) and to Cournot (1843); see Fienberg (2005). In a deterministic universe, based on Newtonian concepts, there would be no probability if all conditions were known ( Laplace's demon ) (but there are situations in which sensitivity to initial conditions exceeds our ability to measure them, i.e. know them). In

6873-419: 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 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

6960-403: The relevant tables it can be established that the type of task in this situation is of the type (F) which is defined as 'Restore or shift a system to original or new state following procedures, with some checking'. This task type has the proposed nominal human unreliability value of 0.003. Other factors to be included in the calculation are provided in the table below: The final calculation for

7047-486: The results that actually occur fall in a given event, the event is said to have occurred. A probability is a 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 for any collection of mutually exclusive events (events with no common results, such as

7134-425: 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 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

7221-496: 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 of behavioral finance emerged to describe

7308-987: The wagering pools, while bookmakers pay off either at the odds offered at the time of accepting the bet; or at the median odds offered by track bookmakers at the time the race started. Betting on team sports has become an important service industry in many countries. Before the advent of the internet, millions of people played the football pools every week in the United Kingdom . In addition to organized sports betting, both legal and illegal, there are many side-betting games played by casual groups of spectators, such as NCAA basketball tournament Bracket Pools, Super Bowl Squares, Fantasy Sports Leagues with monetary entry fees and winnings, and in-person spectator games like Moundball . Based on Sports Betting, Virtual Sports are fantasy and never played sports events made by software that can be played every time without wondering about external things like weather conditions. Arbitrage betting

7395-447: The wave function, believed quantum mechanics is a statistical approximation of an underlying deterministic reality . In some modern interpretations of the statistical mechanics of measurement, quantum decoherence is invoked to account for the appearance of subjectively probabilistic experimental outcomes. Gambling Gambling (also known as betting or gaming ) is the wagering of something of value ("the stakes") on

7482-508: Was developed by Williams in 1986. It is a first generation HRA technique, yet it is dissimilar to many of its contemporaries in that it remains to be widely used throughout the UK . The method essentially takes into consideration all factors which may negatively affect performance of a task in which human reliability is considered to be dependent, and each of these factors is then independently quantified to obtain an overall Human Error Probability (HEP),

7569-465: Was proposed in 1778 by Laplace, and stated that the frequency of the error is an exponential function of the square of the error. 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." Daniel Bernoulli (1778) introduced

#819180