In probability theory , odds provide a measure of the probability of a particular outcome. Odds are commonly used in gambling and statistics . For example for an event that is 40% probable, one could say that the odds are "2 in 5", "2 to 3 in favor", or "3 to 2 against".
131-431: When gambling , odds are often given as the ratio of the possible net profit to the possible net loss. However in many situations, you pay the possible loss ("stake" or "wager") up front and, if you win, you are paid the net win plus you also get your stake returned. So wagering 2 at "3 to 2" , pays out 3 + 2 = 5 , which is called "5 for 2". When Moneyline odds are quoted as a positive number + X , it means that
262-470: A bookmaker , and an individual, rather than between individuals. Different traditions have grown up in how to express odds to customers. Favoured by bookmakers in the United Kingdom and Ireland , and also common in horse racing , fractional odds quote the net total that will be paid out to the bettor, should they win, relative to the stake. Odds of 4/1 would imply that the bettor stands to make
393-597: 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
524-515: A complex fraction , either the numerator, or the denominator, or both, is a fraction or a mixed number, corresponding to division of fractions. For example, 1 / 2 1 / 3 {\displaystyle {\tfrac {1/2}{1/3}}} and ( 12 3 4 ) / 26 {\displaystyle {\bigl (}12{\tfrac {3}{4}}{\bigr )}{\big /}26} are complex fractions. To interpret nested fractions written "stacked" with
655-404: A decimal separator , the appearance of which (e.g., a period, an interpunct (·), a comma) depends on the locale (for examples, see Decimal separator ). Thus, for 0.75 the numerator is 75 and the implied denominator is 10 to the second power, namely, 100, because there are two digits to the right of the decimal separator. In decimal numbers greater than 1 (such as 3.75), the fractional part of
786-448: A last specific event which is solved by the odds algorithm . The odds are a ratio of probabilities; an odds ratio is a ratio of odds, that is, a ratio of ratios of probabilities. Odds-ratios are often used in analysis of clinical trials . While they have useful mathematical properties, they can produce counter- intuitive results: an event with an 80% probability of occurring is four times more probable to happen than an event with
917-441: 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
1048-524: A "/" but can also be represented with a "-", e.g. 4/1 or 4–1. Odds with a denominator of 1 are often presented in listings as the numerator only. A variation of fractional odds is known as Hong Kong odds. Fractional and Hong Kong odds are actually exchangeable. The only difference is that the UK odds are presented as a fractional notation (e.g. 6/5) whilst the Hong Kong odds are decimal (e.g. 1.2). Both exhibit
1179-402: A 20% probability, but the odds are 16 times higher on the less probable event (4–1 against , or 4) than on the more probable one (1–4, or 4–1 on , or 0.25). Answer: The odds in favour of a blue marble are 2:13. One can equivalently say that the odds are 13:2 against . There are 2 out of 15 chances in favour of blue, 13 out of 15 against blue. In probability theory and statistics , where
1310-470: A basic example, two entire cakes and three quarters of another cake might be written as 2 3 4 {\displaystyle 2{\tfrac {3}{4}}} cakes or 2 3 / 4 {\displaystyle 2\ \,3/4} cakes, with the numeral 2 {\displaystyle 2} representing the whole cakes and the fraction 3 4 {\displaystyle {\tfrac {3}{4}}} representing
1441-422: A bookmaker's brokerage fee ("vig" or vigorish ). Also, depending on how the betting is affected by jurisdiction, taxes may be involved for the bookmaker and/or the winning player. This may be taken into account when offering the odds and/or may reduce the amount won by a player. Gambling Gambling (also known as betting or gaming ) is the wagering of something of value ("the stakes") on
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#17327728684151572-430: A cake into four pieces; two of the pieces together ( 2 / 4 ) make up half the cake ( 1 / 2 ). Dividing the numerator and denominator of a fraction by the same non-zero number yields an equivalent fraction: if the numerator and the denominator of a fraction are both divisible by a number (called a factor) greater than 1, then the fraction can be reduced to an equivalent fraction with
1703-492: 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
1834-480: A common fraction. In Unicode, precomposed fraction characters are in the Number Forms block. Common fractions can be classified as either proper or improper. When the numerator and the denominator are both positive, the fraction is called proper if the numerator is less than the denominator, and improper otherwise. The concept of an "improper fraction" is a late development, with the terminology deriving from
1965-407: 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,
2096-662: 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
2227-841: A horizontal fraction bars, treat shorter bars as nested inside longer bars. Complex fractions can be simplified using multiplication by the reciprocal, as described below at § Division . For example: A complex fraction should never be written without an obvious marker showing which fraction is nested inside the other, as such expressions are ambiguous. For example, the expression 5 / 10 / 20 {\displaystyle 5/10/20} could be plausibly interpreted as either 5 10 / 20 = 1 40 {\displaystyle {\tfrac {5}{10}}{\big /}20={\tfrac {1}{40}}} or as 5 / 10 20 = 10. {\displaystyle 5{\big /}{\tfrac {10}{20}}=10.} The meaning can be made explicit by writing
2358-427: A line (or before a slash like 1 ⁄ 2 ), and a non-zero integer denominator , displayed below (or after) that line. If these integers are positive, then the numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. For example, in the fraction 3 / 4 , the numerator 3 indicates that the fraction represents 3 equal parts, and
2489-406: A multi-runner horse race or a football match between two unequally matched teams, betting "at odds" provides the possibility to take the respective likelihoods of the possible outcomes into account. The use of odds in gambling facilitates betting on events where the probabilities of different outcomes vary. In the modern era, most fixed-odd betting takes place between a betting organisation, such as
2620-489: A multiple (if it is equal to or greater than one) of the likelihood that the event will not happen. In the first example at top, saying the odds of a Sunday are "one to six" or, less commonly, "one-sixth" means the probability of picking a Sunday randomly is one-sixth the probability of not picking a Sunday. While the mathematical probability of an event has a value in the range from zero to one, "the odds" in favor of that same event lie between zero and infinity. The odds against
2751-418: A number of "fifths".) Exceptions include the denominator 2, which is always read "half" or "halves", the denominator 4, which may be alternatively expressed as "quarter"/"quarters" or as "fourth"/"fourths", and the denominator 100, which may be alternatively expressed as "hundredth"/"hundredths" or " percent ". When the denominator is 1, it may be expressed in terms of "wholes" but is more commonly ignored, with
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#17327728684152882-587: 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
3013-453: A number) are related by simple formulas, and similarly odds in favor and odds against, and probability of success and probability of failure have simple relations. Odds range from 0 to infinity, while probabilities range from 0 to 1, and hence are often represented as a percentage between 0% and 100%: reversing the ratio switches odds for with odds against, and similarly probability of success with probability of failure. Given odds (in favor) as
3144-405: A person randomly chose one car on the lot, then there is a one in three chance or probability that it would be yellow. A decimal fraction is a fraction whose denominator is not given explicitly, but is understood to be an integer power of ten. Decimal fractions are commonly expressed using decimal notation in which the implied denominator is determined by the number of digits to the right of
3275-409: A piece of type bearing a complete fraction (e.g. 1 / 2 ) was known as a "case fraction", while those representing only part of fraction were called "piece fractions". The denominators of English fractions are generally expressed as ordinal numbers , in the plural if the numerator is not 1. (For example, 2 / 5 and 3 / 5 are both read as
3406-401: A ratio of two numbers, in which case it is not unique—scaling both terms by the same factor does not change the proportions: 1:1 odds and 100:100 odds are the same (even odds). Odds can also be expressed as a number, by dividing the terms in the ratio—in this case it is unique (different fractions can represent the same rational number ). Odds as a ratio, odds as a number, and probability (also
3537-436: A smaller numerator and a smaller denominator. For example, if both the numerator and the denominator of the fraction a b {\displaystyle {\tfrac {a}{b}}} are divisible by c {\displaystyle c} , then they can be written as a = c d {\displaystyle a=cd} , b = c e {\displaystyle b=ce} , and
3668-769: 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
3799-430: 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
3930-534: A sum of unit fractions in infinitely many ways. Two ways to write 13 17 {\displaystyle {\tfrac {13}{17}}} are 1 2 + 1 4 + 1 68 {\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{4}}+{\tfrac {1}{68}}} and 1 3 + 1 4 + 1 6 + 1 68 {\displaystyle {\tfrac {1}{3}}+{\tfrac {1}{4}}+{\tfrac {1}{6}}+{\tfrac {1}{68}}} . In
4061-498: 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
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4192-469: A wager pays X to 100. When Moneyline odds are quoted as a negative number − X , it means that a wager pays 100 to X . Odds have a simple relationship with probability . When probability is expressed as a number between 0 and 1, the relationships between probability p and odds are as follows. Note that if probability is to be expressed as a percentage these probability values should be multiplied by 100%. The numbers for odds can be scaled. If k
4323-472: A whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common , vulgar , or simple fraction (examples: 1 2 {\displaystyle {\tfrac {1}{2}}} and 17 3 {\displaystyle {\tfrac {17}{3}}} ) consists of an integer numerator , displayed above
4454-421: A £400 profit on a £100 stake. If the odds are 1/4, the bettor will make £25 on a £100 stake. In either case, having won, the bettor always receives the original stake back; so if the odds are 4/1 the bettor receives a total of £500 (£400 plus the original £100). Odds of 1/1 are known as evens or even money . The numerator and denominator of fractional odds are often integers , thus if the bookmaker's payout
4585-432: Is in lowest terms—the only positive integer that goes into both 3 and 8 evenly is 1. Using these rules, we can show that 5 / 10 = 1 / 2 = 10 / 20 = 50 / 100 , for example. As another example, since the greatest common divisor of 63 and 462 is 21, the fraction 63 / 462 can be reduced to lowest terms by dividing
4716-402: Is 1, hence the reciprocal is the multiplicative inverse of a fraction. The reciprocal of a proper fraction is improper, and the reciprocal of an improper fraction not equal to 1 (that is, numerator and denominator are not equal) is a proper fraction. When the numerator and denominator of a fraction are equal (for example, 7 / 7 ), its value is 1, and the fraction therefore
4847-415: Is 100%, thus representing a fair 'book'. The true odds against winning for each of the three horses are 1–1, 3–2 and 9–1, respectively. In order to generate a profit on the wagers accepted, the bookmaker may decide to increase the values to 60%, 50% and 20% for the three horses, respectively. This represents the odds against each, which are 4–6, 1–1 and 4–1, in order. These values now total 130%, meaning that
4978-403: Is 4 to 2 and may be expressed as 4:2 or 2:1. A ratio is often converted to a fraction when it is expressed as a ratio to the whole. In the above example, the ratio of yellow cars to all the cars on the lot is 4:12 or 1:3. We can convert these ratios to a fraction, and say that 4 / 12 of the cars or 1 / 3 of the cars in the lot are yellow. Therefore, if
5109-407: Is 75/1,000,000. Whether common fractions or decimal fractions are used is often a matter of taste and context. Common fractions are used most often when the denominator is relatively small. By mental calculation , it is easier to multiply 16 by 3/16 than to do the same calculation using the fraction's decimal equivalent (0.1875). And it is more accurate to multiply 15 by 1/3, for example, than it
5240-407: Is a Bernoulli trial , as it has exactly two outcomes. In case of a finite sample space of equally probable outcomes , this is the ratio of the number of outcomes where the event occurs to the number of outcomes where the event does not occur; these can be represented as W and L (for Wins and Losses) or S and F (for Success and Failure). For example, the odds that a randomly chosen day of
5371-427: Is a parabolic transform . In probability theory and statistics, odds and similar ratios may be more natural or more convenient than probabilities. In some cases the log-odds are used, which is the logit of the probability. Most simply, odds are frequently multiplied or divided, and log converts multiplication to addition and division to subtractions. This is particularly important in the logistic model , in which
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5502-591: 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
5633-881: Is a minor difference if the probability is small (close to zero, or "long odds"), but is a major difference if the probability is large (close to one). These are worked out for some simple odds: These transforms have certain special geometric properties: the conversions between odds for and odds against (resp. probability of success with probability of failure) and between odds and probability are all Möbius transformations (fractional linear transformations). They are thus specified by three points ( sharply 3-transitive ). Swapping odds for and odds against swaps 0 and infinity, fixing 1, while swapping probability of success with probability of failure swaps 0 and 1, fixing .5; these are both order 2, hence circular transforms . Converting odds to probability fixes 0, sends infinity to 1, and sends 1 to .5 (even odds are 50% probable), and conversely; this
5764-436: Is a relationship between two or more numbers that can be sometimes expressed as a fraction. Typically, a number of items are grouped and compared in a ratio, specifying numerically the relationship between each group. Ratios are expressed as "group 1 to group 2 ... to group n ". For example, if a car lot had 12 vehicles, of which then the ratio of red to white to yellow cars is 6 to 2 to 4. The ratio of yellow cars to white cars
5895-464: 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
6026-543: 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
6157-822: Is any positive number then X to Y is the same as kX to kY , and similarly if "to" is replaced with "in" or "for". For example, "3 to 2 against" is the same as both "1.5 to 1 against" and "6 to 4 against". When the value of the probability p (between 0 and 1; not a percentage) can be written as a fraction N / D then the odds can be said to be " p /(1− p ) to 1 in favor", " (1− p )/ p to 1 against", " N in D ", " N to D − N in favor", or " D − N to N against", and these can be scaled to equivalent odds. Similarly, fair betting odds can be expressed as " (1− p )/ p to 1", " 1/ p for 1", "+ 100(1− p )/ p ", " −100 p /(1− p ) ", " D − N to N ", " D for N ", "+ 100( D − N )/ N ", or " −100 N /( D − N ) ". The language of odds, such as
6288-443: Is improper. Its reciprocal is identical and hence also equal to 1 and improper. Any integer can be written as a fraction with the number one as denominator. For example, 17 can be written as 17 / 1 , where 1 is sometimes referred to as the invisible denominator . Therefore, every fraction or integer, except for zero, has a reciprocal. For example, the reciprocal of 17 is 1 / 17 . A ratio
6419-408: Is known as the 'overround' on the 'book' (the 'book' refers to the old-fashioned ledger in which wagers were recorded, and is the derivation of the term 'bookmaker') and relates to the sum of the 'odds' in the following way: In a 3-horse race, for example, the true probabilities of each of the horses winning based on their relative abilities may be 50%, 40% and 10%. The total of these three percentages
6550-623: 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
6681-695: 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
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#17327728684156812-475: Is of the type named "fifth". In terms of division , the numerator corresponds to the dividend , and the denominator corresponds to the divisor . Informally, the numerator and denominator may be distinguished by placement alone, but in formal contexts they are usually separated by a fraction bar . The fraction bar may be horizontal (as in 1 / 3 ), oblique (as in 2/5), or diagonal (as in 4 ⁄ 9 ). These marks are respectively known as
6943-508: 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
7074-425: Is the fact that the probability of an event is given by the ratio of favourable outcomes to the total number of possible outcomes. In statistics, odds are an expression of relative probabilities, generally quoted as the odds in favor . The odds (in favor) of an event or a proposition is the ratio of the probability that the event will happen to the probability that the event will not happen. Mathematically, this
7205-465: Is the fraction 2 / 5 and "two fifths" is the same fraction understood as 2 instances of 1 / 5 .) Fractions should always be hyphenated when used as adjectives. Alternatively, a fraction may be described by reading it out as the numerator "over" the denominator, with the denominator expressed as a cardinal number . (For example, 3 / 1 may also be expressed as "three over one".) The term "over"
7336-461: Is the same as multiplying by one, and any number multiplied by one has the same value as the original number. By way of an example, start with the fraction 1 2 {\displaystyle {\tfrac {1}{2}}} . When the numerator and denominator are both multiplied by 2, the result is 2 / 4 , which has the same value (0.5) as 1 / 2 . To picture this visually, imagine cutting
7467-506: Is to find a common denominator. To compare a b {\displaystyle {\tfrac {a}{b}}} and c d {\displaystyle {\tfrac {c}{d}}} , these are converted to a ⋅ d b ⋅ d {\displaystyle {\tfrac {a\cdot d}{b\cdot d}}} and b ⋅ c b ⋅ d {\displaystyle {\tfrac {b\cdot c}{b\cdot d}}} (where
7598-446: Is to multiply 15 by any decimal approximation of one third. Monetary values are commonly expressed as decimal fractions with denominator 100, i.e., with two decimals, for example $ 3.75. However, as noted above, in pre-decimal British currency, shillings and pence were often given the form (but not the meaning) of a fraction, as, for example, "3/6" (read "three and six") meaning 3 shillings and 6 pence, and having no relationship to
7729-633: Is used as a synonym for the other. (For example, the compound fraction 3 4 × 5 7 {\displaystyle {\tfrac {3}{4}}\times {\tfrac {5}{7}}} is equivalent to the complex fraction 3 / 4 7 / 5 {\displaystyle {\tfrac {3/4}{7/5}}} .) Nevertheless, "complex fraction" and "compound fraction" may both be considered outdated and now used in no well-defined manner, partly even taken synonymously for each other or for mixed numerals. They have lost their meaning as technical terms and
7860-426: Is used even in the case of solidus fractions, where the numbers are placed left and right of a slash mark . (For example, 1/2 may be read "one-half", "one half", or "one over two".) Fractions with large denominators that are not powers of ten are often rendered in this fashion (e.g., 1 / 117 as "one over one hundred seventeen"), while those with denominators divisible by ten are typically read in
7991-1058: 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
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#17327728684158122-468: The rational fraction 1 x {\displaystyle \textstyle {\frac {1}{x}}} ). In a fraction, the number of equal parts being described is the numerator (from Latin : numerātor , "counter" or "numberer"), and the type or variety of the parts is the denominator (from Latin : dēnōminātor , "thing that names or designates"). As an example, the fraction 8 / 5 amounts to eight parts, each of which
8253-607: 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 ),
8384-433: The "real odds" or 100% probability of an event occurring. This 100% book is displayed without any bookmaker 's profit margin , often referred to as a bookmaker's " overround " built in. A "wholesale odds" index is an index of all the prices in a probabilistic market operating at 100% competitiveness and displayed without any profit margin factored for market participants. In gambling, the odds on display do not represent
8515-641: 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
8646-431: The absolute value of the fraction is greater than or equal to 1. Examples of proper fractions are 2/3, −3/4, and 4/9, whereas examples of improper fractions are 9/4, −4/3, and 3/3. The reciprocal of a fraction is another fraction with the numerator and denominator exchanged. The reciprocal of 3 / 7 , for instance, is 7 / 3 . The product of a non-zero fraction and its reciprocal
8777-634: 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
8908-744: The additional partial cake juxtaposed; this is more concise than the more explicit notation 2 + 3 4 {\displaystyle 2+{\tfrac {3}{4}}} cakes. The mixed number 2 + 3 / 4 is pronounced "two and three quarters", with the integer and fraction portions connected by the word and . Subtraction or negation is applied to the entire mixed numeral, so − 2 3 4 {\displaystyle -2{\tfrac {3}{4}}} means − ( 2 + 3 4 ) . {\displaystyle -{\bigl (}2+{\tfrac {3}{4}}{\bigr )}.} Any mixed number can be converted to an improper fraction by applying
9039-713: 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
9170-413: 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
9301-486: 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
9432-428: The attributes "complex" and "compound" tend to be used in their every day meaning of "consisting of parts". Like whole numbers, fractions obey the commutative , associative , and distributive laws, and the rule against division by zero . Mixed-number arithmetic can be performed either by converting each mixed number to an improper fraction, or by treating each as a sum of integer and fractional parts. Multiplying
9563-430: The book has an overround of 30 (130−100). This value of 30 represents the amount of profit for the bookmaker if he gets bets in good proportions on each of the horses. For example, if he takes £60, £50, and £20 of stakes, respectively, for the three horses, he receives £130 in wagers but only pays £100 back (including stakes), whichever horse wins. And the expected value of his profit is positive even if everybody bets on
9694-1037: The convention that juxtaposition in algebraic expressions means multiplication. An Egyptian fraction is the sum of distinct positive unit fractions, for example 1 2 + 1 3 {\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{3}}} . This definition derives from the fact that the ancient Egyptians expressed all fractions except 1 2 {\displaystyle {\tfrac {1}{2}}} , 2 3 {\displaystyle {\tfrac {2}{3}}} and 3 4 {\displaystyle {\tfrac {3}{4}}} in this manner. Every positive rational number can be expanded as an Egyptian fraction. For example, 5 7 {\displaystyle {\tfrac {5}{7}}} can be written as 1 2 + 1 6 + 1 21 . {\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{6}}+{\tfrac {1}{21}}.} Any positive rational number can be written as
9825-415: The decimal point 7 places to the left. Decimal fractions with infinitely many digits to the right of the decimal separator represent an infinite series . For example, 1 / 3 = 0.333... represents the infinite series 3/10 + 3/100 + 3/1000 + .... Another kind of fraction is the percentage (from Latin : per centum , meaning "per hundred", represented by the symbol %), in which
9956-400: The decimal value of the fractional odds plus one. Thus even odds 1/1 are quoted in decimal odds as 2.00. The 4/1 fractional odds discussed above are quoted as 5.00, while the 1/4 odds are quoted as 1.25. This is considered to be ideal for parlay betting, because the odds to be paid out are simply the product of the odds for each outcome wagered on. When looking at decimal odds in betting terms,
10087-525: The decimalized metric system . However, scientific measurements typically use the metric system, which is based on decimal fractions, and starting from the secondary school level, mathematics pedagogy treats every fraction uniformly as a rational number , the quotient p / q of integers, leaving behind the concepts of "improper fraction" and "mixed number". College students with years of mathematical training are sometimes confused when re-encountering mixed numbers because they are used to
10218-555: The denominator ( b ) cannot be zero. Examples include 1 / 2 , − 8 / 5 , −8 / 5 , and 8 / −5 . The term was originally used to distinguish this type of fraction from the sexagesimal fraction used in astronomy. Common fractions can be positive or negative, and they can be proper or improper (see below). Compound fractions, complex fractions, mixed numerals, and decimals (see below) are not common fractions ; though, unless irrational, they can be evaluated to
10349-427: The denominator 4 indicates that 4 parts make up a whole. The picture to the right illustrates 3 / 4 of a cake. Fractions can be used to represent ratios and division . Thus the fraction 3 / 4 can be used to represent the ratio 3:4 (the ratio of the part to the whole), and the division 3 ÷ 4 (three divided by four). We can also write negative fractions, which represent
10480-516: 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
10611-750: 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 ,
10742-621: The dot signifies multiplication and is an alternative symbol to ×). Then bd is a common denominator and the numerators ad and bc can be compared. It is not necessary to determine the value of the common denominator to compare fractions – one can just compare ad and bc , without evaluating bd , e.g., comparing 2 3 {\displaystyle {\tfrac {2}{3}}} ? 1 2 {\displaystyle {\tfrac {1}{2}}} gives 4 6 > 3 6 {\displaystyle {\tfrac {4}{6}}>{\tfrac {3}{6}}} . For
10873-572: 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
11004-428: The event with probability given as p are 1 − p p {\displaystyle {\frac {1-p}{p}}} . The odds against Sunday are 6:1 or 6/1 = 6. It is 6 times as probable that a random day is not a Sunday. On a coin toss or a match race between two evenly matched horses, it is reasonable for two people to wager level stakes. However, in more variable situations, such as
11135-434: The fact that "fraction" means "a piece", so a proper fraction must be less than 1. This was explained in the 17th century textbook The Ground of Arts . In general, a common fraction is said to be a proper fraction , if the absolute value of the fraction is strictly less than one—that is, if the fraction is greater than −1 and less than 1. It is said to be an improper fraction , or sometimes top-heavy fraction , if
11266-408: The fraction 3/6. A mixed number (also called a mixed fraction or mixed numeral ) is the sum of a non-zero integer and a proper fraction, conventionally written by juxtaposition (or concatenation ) of the two parts, without the use of an intermediate plus (+) or minus (−) sign. When the fraction is written horizontally, a space is added between the integer and fraction to separate them. As
11397-399: The fraction becomes cd / ce , which can be reduced by dividing both the numerator and denominator by c to give the reduced fraction d / e . If one takes for c the greatest common divisor of the numerator and the denominator, one gets the equivalent fraction whose numerator and denominator have the lowest absolute values . One says that
11528-504: The fraction has been reduced to its lowest terms . If the numerator and the denominator do not share any factor greater than 1, the fraction is already reduced to its lowest terms, and it is said to be irreducible , reduced , or in simplest terms . For example, 3 9 {\displaystyle {\tfrac {3}{9}}} is not in lowest terms because both 3 and 9 can be exactly divided by 3. In contrast, 3 8 {\displaystyle {\tfrac {3}{8}}}
11659-405: The fractions using distinct separators or by adding explicit parentheses, in this instance ( 5 / 10 ) / 20 {\displaystyle (5/10){\big /}20} or 5 / ( 10 / 20 ) . {\displaystyle 5{\big /}(10/20).} A compound fraction is a fraction of a fraction, or any number of fractions connected with
11790-401: The greater the amount of gambling activity in a given country, the more volatile that country's stock-market prices are. Legalization of online sports betting was found to decrease household saving, decrease investment with positive expected value and increase financial distress . Fraction (mathematics) A fraction (from Latin : fractus , "broken") represents a part of
11921-472: 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
12052-445: The horizontal bar; the virgule, slash ( US ), or stroke ( UK ); and the fraction bar, solidus, or fraction slash . In typography , fractions stacked vertically are also known as " en " or " nut fractions", and diagonal ones as " em " or "mutton fractions", based on whether a fraction with a single-digit numerator and denominator occupies the proportion of a narrow en square, or a wider em square. In traditional typefounding ,
12183-400: The implied denominator is always 100. Thus, 51% means 51/100. Percentages greater than 100 or less than zero are treated in the same way, e.g. 311% equals 311/100, and −27% equals −27/100. The related concept of permille or parts per thousand (ppt) has an implied denominator of 1000, while the more general parts-per notation , as in 75 parts per million (ppm), means that the proportion
12314-427: The inverse of the probability of an outcome. For example, a quoted odds of 5.00 equals to a probability of 1 / 5.00, that is 0.20 or 20%. Decimal odds are also known as European odds , digital odds or continental odds. Moneyline odds are favoured by American bookmakers. The figure quoted is either positive or negative. Moneyline odds are often referred to as American odds . A "moneyline" wager refers to odds on
12445-516: The log-odds of the target variable are a linear combination of the observed variables. Similar ratios are used elsewhere in statistics; of central importance is the likelihood ratio in likelihoodist statistics , which is used in Bayesian statistics as the Bayes factor . Odds are particularly useful in problems of sequential decision making, as for instance in problems of how to stop (online) on
12576-559: 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
12707-616: The more laborious question 5 18 {\displaystyle {\tfrac {5}{18}}} ? 4 17 , {\displaystyle {\tfrac {4}{17}},} multiply top and bottom of each fraction by the denominator of the other fraction, to get a common denominator, yielding 5 × 17 18 × 17 {\displaystyle {\tfrac {5\times 17}{18\times 17}}} ? 18 × 4 18 × 17 {\displaystyle {\tfrac {18\times 4}{18\times 17}}} . It
12838-416: The net return. The European odds also represent the potential winnings (net returns), but in addition they factor in the stake (e.g. 6/5 or 1.2 plus 1 = 2.2). Favoured in continental Europe , Australia , New Zealand , Canada , and Singapore , decimal odds quote the ratio of the payout amount, including the original stake, to the stake itself. Therefore, the decimal odds of an outcome are equivalent to
12969-452: The normal ordinal fashion (e.g., 6 / 1000000 as "six-millionths", "six millionths", or "six one-millionths"). A simple fraction (also known as a common fraction or vulgar fraction , where vulgar is Latin for "common") is a rational number written as a / b or a b {\displaystyle {\tfrac {a}{b}}} , where a and b are both integers . As with other fractions,
13100-419: The number for and against combined. For example, "odds of a weekend are 2 to 5", while "chances of a weekend are 2 in 7". In casual use, the words odds and chances (or chance ) are often used interchangeably to vaguely indicate some measure of odds or probability, though the intended meaning can be deduced by noting whether the preposition between the two numbers is to or in . Odds can be expressed as
13231-440: The number is expressed by the digits to the right of the decimal (with a value of 0.75 in this case). 3.75 can be written either as an improper fraction, 375/100, or as a mixed number, 3 + 75 / 100 . Decimal fractions can also be expressed using scientific notation with negative exponents, such as 6.023 × 10 , which represents 0.0000006023. The 10 represents a denominator of 10 . Dividing by 10 moves
13362-402: The numerator and denominator by 21: The Euclidean algorithm gives a method for finding the greatest common divisor of any two integers. Comparing fractions with the same positive denominator yields the same result as comparing the numerators: If the equal denominators are negative, then the opposite result of comparing the numerators holds for the fractions: If two positive fractions have
13493-427: The numerator and denominator of a fraction by the same (non-zero) number results in a fraction that is equivalent to the original fraction. This is true because for any non-zero number n {\displaystyle n} , the fraction n n {\displaystyle {\tfrac {n}{n}}} equals 1. Therefore, multiplying by n n {\displaystyle {\tfrac {n}{n}}}
13624-425: The numerator read out as a whole number. For example, 3 / 1 may be described as "three wholes", or simply as "three". When the numerator is 1, it may be omitted (as in "a tenth" or "each quarter"). The entire fraction may be expressed as a single composition, in which case it is hyphenated, or as a number of fractions with a numerator of one, in which case they are not. (For example, "two-fifths"
13755-421: The odds (for example, 46.6% for even odds). It was about 3.7% less for wins by the visitors, and 5.7% less for draws. To understand roulette probabilities and calculate them, you need to know the formula. You take the numbers your bet is on and divide them by the total number of numbers in roulette (depending on your version of the game). Then you multiply by 100. Making a profit in gambling involves predicting
13886-451: The odds as numbers can be computed by dividing: Conversely, given the odds as a number o f , {\displaystyle o_{f},} this can be represented as the ratio o f : 1 , {\displaystyle o_{f}:1,} or conversely 1 : ( 1 / o f ) = 1 : o a , {\displaystyle 1:(1/o_{f})=1:o_{a},} from which
14017-509: 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
14148-410: The opposite of a positive fraction. For example, if 1 / 2 represents a half-dollar profit, then − 1 / 2 represents a half-dollar loss. Because of the rules of division of signed numbers (which states in part that negative divided by positive is negative), − 1 / 2 , −1 / 2 and 1 / −2 all represent
14279-538: 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
14410-605: 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
14541-401: 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
14672-448: The probability of success and probability of failure sum to unity (one), as they are the only possible outcomes. In case of a finite number of equally probable outcomes, this can be interpreted as the number of outcomes where the event occurs divided by the total number of events: Given a probability p, the odds as a ratio is p : q {\displaystyle p:q} (probability of success to probability of failure), and
14803-399: The probability of success or failure can be computed: Thus if expressed as a fraction with a numerator of 1, probability and odds differ by exactly 1 in the denominator: a probability of 1 in 100 (1/100 = 1%) is the same as odds of 1 to 99 (1/99 = 0.0101... = 0. 01 ), while odds of 1 to 100 (1/100 = 0.01) is the same as a probability of 1 in 101 (1/101 = 0.00990099... = 0. 0099 ). This
14934-461: The ratio W:L (number of outcomes that are wins:number of outcomes that are losses), the odds in favor (as a number) o f {\displaystyle o_{f}} and odds against (as a number) o a {\displaystyle o_{a}} can be computed by simply dividing, and are multiplicative inverses : Analogously, given odds as a ratio, the probability of success p or failure q can be computed by dividing, and
15065-741: The ratio by the number of outcomes yields the probabilities: 2 : 5 = ( 2 / 7 ) : ( 5 / 7 ) . {\displaystyle 2:5=(2/7):(5/7).} Conversely, the odds against is the opposite ratio. For example, the odds against a random day of the week being during a weekend are 5:2. Odds and probability can be expressed in prose via the prepositions to and in: "odds of so many to so many on (or against) [some event]" refers to odds —the ratio of numbers of (equally probable) outcomes in favor and against (or vice versa); "chances of so many [outcomes], in so many [outcomes]" refers to probability —the number of (equally probable) outcomes in favour relative to
15196-409: The relationship of the true probabilities to the payout odds. Sports information services are often used by professional and semi-professional sports bettors to help achieve this goal. The odds or amounts the bookmaker will pay are determined by the total amount that has been bet on all of the possible events. They reflect the balance of wagers on either side of the event, and include the deduction of
15327-591: The remainder divided by the divisor. For example, since 4 goes into 11 twice, with 3 left over, 11 4 = 2 + 3 4 . {\displaystyle {\tfrac {11}{4}}=2+{\tfrac {3}{4}}.} In primary school, teachers often insist that every fractional result should be expressed as a mixed number. Outside school, mixed numbers are commonly used for describing measurements, for instance 2 + 1 / 2 hours or 5 3/16 inches , and remain widespread in daily life and in trades, especially in regions that do not use
15458-419: The rules of adding unlike quantities . For example, 2 + 3 4 = 8 4 + 3 4 = 11 4 . {\displaystyle 2+{\tfrac {3}{4}}={\tfrac {8}{4}}+{\tfrac {3}{4}}={\tfrac {11}{4}}.} Conversely, an improper fraction can be converted to a mixed number using division with remainder , with the proper fraction consisting of
15589-425: The same fraction – negative one-half. And because a negative divided by a negative produces a positive, −1 / −2 represents positive one-half. In mathematics a rational number is a number that can be represented by a fraction of the form a / b , where a and b are integers and b is not zero; the set of all rational numbers is commonly represented by
15720-409: The same horse. The art of bookmaking is in setting the odds low enough so as to have a positive expected value of profit while keeping the odds high enough to attract customers, and at the same time attracting enough bets for each outcome to reduce his risk exposure. A study on soccer betting found that the probability for the home team to win was generally about 3.4% less than the value calculated from
15851-458: The same numerator, then the fraction with the smaller denominator is the larger number. When a whole is divided into equal pieces, if fewer equal pieces are needed to make up the whole, then each piece must be larger. When two positive fractions have the same numerator, they represent the same number of parts, but in the fraction with the smaller denominator, the parts are larger. One way to compare fractions with different numerators and denominators
15982-418: The straight-up outcome of a game with no consideration to a point spread . In most cases, the favorite will have negative moneyline odds (less payoff for a safer bet) and the underdog will have positive moneyline odds (more payoff for a risky bet). However, if the teams are evenly matched, both teams can have a negative line at the same time (e.g. −110 −110 or −105 −115), due to house take. Wholesale odds are
16113-444: The symbol Q or Q {\displaystyle \mathbb {Q} } , which stands for quotient . The term fraction and the notation a / b can also be used for mathematical expressions that do not represent a rational number (for example 2 2 {\displaystyle \textstyle {\frac {\sqrt {2}}{2}}} ), and even do not represent any number (for example
16244-425: The true chances (as imagined by the bookmaker) that the event will or will not occur, but are the amount that the bookmaker will pay out on a winning bet, together with the required stake. In formulating the odds to display the bookmaker will have included a profit margin which effectively means that the payout to a successful bettor is less than that represented by the true chance of the event occurring. This profit
16375-451: The underdog has the higher of the two decimals, while the favorite has the lower of the two. To calculate decimal odds, you can use the equation Payout = Initial Wager × Decimal Value . For example, if you bet €100 on Liverpool to beat Manchester City at 2.00 odds the payout, including your stake, would be €200 (€100 × 2.00). Decimal odds are favoured by betting exchanges because they are the easiest to work with for trading, as they reflect
16506-441: The use of phrases like "ten to one" for intuitively estimated risks, is found in the sixteenth century, well before the development of probability theory . Shakespeare wrote: Knew that we ventured on such dangerous seas That if we wrought out life 'twas ten to one The sixteenth-century polymath Cardano demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes. Implied by this definition
16637-405: The variable p is the probability in favor of a binary event, and the probability against the event is therefore 1- p , "the odds" of the event are the quotient of the two, or p 1 − p {\displaystyle {\frac {p}{1-p}}} . That value may be regarded as the relative probability the event will happen, expressed as a fraction (if it is less than 1), or
16768-931: 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
16899-412: The week is during a weekend are two to five (2:5), as days of the week form a sample space of seven outcomes, and the event occurs for two of the outcomes (Saturday and Sunday), and not for the other five. Conversely, given odds as a ratio of integers, this can be represented by a probability space of a finite number of equally probable outcomes. These definitions are equivalent, since dividing both terms in
17030-661: The word of , corresponding to multiplication of fractions. To reduce a compound fraction to a simple fraction, just carry out the multiplication (see § Multiplication ). For example, 3 4 {\displaystyle {\tfrac {3}{4}}} of 5 7 {\displaystyle {\tfrac {5}{7}}} is a compound fraction, corresponding to 3 4 × 5 7 = 15 28 {\displaystyle {\tfrac {3}{4}}\times {\tfrac {5}{7}}={\tfrac {15}{28}}} . The terms compound fraction and complex fraction are closely related and sometimes one
17161-551: Was to be £1.25 for every £1 stake, this would be equivalent to £5 for every £4 staked, and the odds would therefore be expressed as 5/4. However, not all fractional odds are traditionally read using the lowest common denominator . For example, given that there is a pattern of odds of 5/4, 7/4, 9/4 and so on, odds which are mathematically 3/2 are more easily compared if expressed in the equivalent form 6/4. Fractional odds are also known as British odds, UK odds, or, in that country, traditional odds . They are typically represented with
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