Inprobability theory,odds provide a measure of the probability of a particular outcome. Odds are commonly used ingambling andstatistics. For example for an event that is 40% probable, one could say that the odds are"2 in 5","2 to 3 in favor","2 to 3 on", or"3 to 2 against".
Ingambling, odds are often given as the ratio of the possible net profitto the possible net loss. However in many situations, the possible loss ("stake" or "wager") is paid up front and, if the gambler wins, the net win plus the stake is returned. So wagering 2 at"3to 2", pays out3 + 2 = 5, which is called"5for 2". WhenMoneyline odds are quoted as a positive number+X, it means that a wager paysX to 100. When Moneyline odds are quoted as a negative number−X, it means that a wager pays100 toX.
Odds have a simple relationship withprobability. When probability is expressed as a number between 0 and 1, the relationships between probabilityp and odds are as follows. Multiplying these probabilities by 100% gives the percentage.
The numbers for odds can be scaled. Ifk is any positive number thenX toY is the same askX tokY, 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 probabilityp (between 0 and 1; not a percentage) can be written as a fractionN /D then the odds can be said to be"p/(1−p) to 1 in favor","(1−p)/p to 1 against","N inD","N toD−N in favor", or"D−N toN 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","−100p/(1−p)","D−N toN","D forN","+100(D−N)/N", or"−100N/(D−N)".
The language of odds, such as the use of phrases like "ten to one" forintuitively estimated risks, is found in the sixteenth century, well before the development ofprobability theory.[1]Shakespeare wrote:
Knew that we ventured on such dangerous seas
That if we wrought out life 'twas ten to one
— William Shakespeare,Henry IV, Part II, Act I, Scene 1, lines 181–2
The sixteenth-centurypolymathCardano demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes. Implied by this definition is the fact that the probability of an event is given by theratio of favourable outcomes to the total number of possible outcomes.[2]
In statistics, odds are an expression of relative probabilities, generally quoted as the oddsin favor. The odds (in favor) of anevent or aproposition is the ratio of the probability that the event will happen to the probability that the event will not happen. Mathematically, this is aBernoulli trial, as it has exactly two outcomes. In case of a finitesample space ofequally probable outcomes, this is the ratio of the number ofoutcomes where the event occurs to the number of outcomes where the event does not occur; these can be represented asW andL (for Wins and Losses) orS andF (for Success and Failure). For example, the odds that arandomly chosen day of 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.[3][4] 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 the ratio by the number of outcomes yields the probabilities: 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 prepositionsto andin: "odds of so manyto so many on (or against) [some event]" refers toodds—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 toprobability—the number of (equally probable) outcomes in favour relative to the number for and against combined. For example, "odds of a weekend are 2to 5", while "chances of a weekend are 2in 7". In casual use, the wordsodds andchances (orchance) 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 isto orin.[5][6][7]
Odds can be expressed as 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 (differentfractions can represent the samerational number). Odds as a ratio, odds as a number, and probability (also 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 the ratio W:L (number of outcomes that are wins:number of outcomes that are losses), the odds in favor (as a number) and odds against (as a number) can be computed by simply dividing, and aremultiplicative inverses:
Analogously, given odds as a ratio, the probability of successp or failureq can be computed by dividing, and the probability of success and probability of failure sum tounity (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 probabilityp, the odds as a ratio is (probability of success to probability of failure), and the odds as numbers can be computed by dividing:
Conversely, given the odds as a number this can be represented as the ratio or conversely from which 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 1in 100 (1/100 = 1%) is the same as odds of 1to 99 (1/99 = 0.0101... = 0.01), while odds of 1to 100 (1/100 = 0.01) is the same as a probability of 1in 101 (1/101 = 0.00990099... = 0.0099). This 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:
| odds (ratio) | ||||
|---|---|---|---|---|
| 1:1 | 1 | 1 | 50% | 50% |
| 0:1 | 0 | ∞ | 0% | 100% |
| 1:0 | ∞ | 0 | 100% | 0% |
| 2:1 | 2 | 0.5 | 66.66% | 33.33% |
| 1:2 | 0.5 | 2 | 33.33% | 66.66% |
| 4:1 | 4 | 0.25 | 80% | 20% |
| 1:4 | 0.25 | 4 | 20% | 80% |
| 9:1 | 9 | 0.1 | 90% | 10% |
| 10:1 | 10 | 0.1 | 90.90% | 9.09% |
| 99:1 | 99 | 0.01 | 99% | 1% |
| 100:1 | 100 | 0.01 | 99.0099% | 0.9900% |
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 allMöbius transformations (fractional linear transformations). They are thusspecified 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, hencecircular transforms. Converting odds to probability fixes 0, sends infinity to 1, and sends 1 to .5 (even odds are 50% probable), and conversely; this is aparabolic transform.
Inprobability theory and statistics, odds and similar ratios may be more natural or more convenient than probabilities. In some cases thelog-odds are used, which is thelogit 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 thelogistic model, in which the log-odds of the target variable are alinear combination of the observed variables.
Similar ratios are used elsewhere in statistics; of central importance is thelikelihood ratio inlikelihoodist statistics, which is used inBayesian statistics as theBayes factor.
Odds are particularly useful in problems of sequential decision making, as for instance in problems of how to stop (online) on alast specific event which is solved by theodds algorithm.
The odds are aratio of probabilities; anodds ratio is a ratio of odds, that is, a ratio of ratios of probabilities. Odds-ratios are often used in analysis ofclinical trials. While they have useful mathematical properties, they can produce counter-intuitive results: an event with an 80% probability of occurring is four timesmore probable to happen than an event with a 20% probability, but theodds are 16 times higher on the less probable event (4–1against, or 4) than on the more probable one (1–4against, 4-1in favor, 4–1on, or 0.25).
Answer: The odds in favour of a blue marble are 2:13. One can equivalently say that the odds are 13:2against. There are 2 out of 15 chances in favour of blue, 13 out of 15 against blue.
Inprobability theory andstatistics, where the variablep is theprobability 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. That value may be regarded as the relative probability the event will happen, expressed as a fraction (if it is less than 1), or 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 the event with probability given asp are. 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 acoin toss or amatch race between two evenly matched horses, it is reasonable for two people to wager level stakes. However, in more variable situations, such as 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 abookmaker, and an individual, rather than between individuals. Different traditions have grown up in how to express odds to customers.
Favoured bybookmakers in theUnited Kingdom andIreland, and also common inhorse racing, fractional odds quote the net total that will be paid out to the bettor, should they win, relative to the stake.[8] Odds of 4/1 (4 to 1 against) would imply that the bettor stands to make a £400 profit on a £100 stake. If the odds are 1/4 (1 to 4 against,4 to 1 in favor, or4 to 1 on), 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 asevens oreven money.
Thenumerator anddenominator of fractional odds are oftenintegers, thus if the bookmaker's payout 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 thelowest 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 asBritish odds,UK odds,[9] or, in that country,traditional odds. They are typically represented with 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.[citation needed]
A variation of fractional odds is known asHong 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 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).[10]
Favoured in continentalEurope,Australia,New Zealand,Canada, andSingapore, 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 the decimal value of the fractional odds plus one.[11] 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 forparlay 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, the underdog has the higher of the two decimals, while the favorite has the lower of the two. The equation for calculating decimal odds isPayout = Initial Wager × Decimal Value[12]. For example, if one bets €100 on Liverpool to beat Manchester City at 2.00 odds, the payout, including the stake, would be €200 (€100 × 2.00). Decimal odds are favoured bybetting exchanges because they are the easiest to work with for trading, as they reflect the reciprocal of the probability of an outcome.[13] 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 asEuropean odds,digital odds orcontinental odds.[9]
Moneyline odds are favoured by American bookmakers. The figure quoted is either positive or negative.
Moneyline odds are often referred to asAmerican odds. A "moneyline" wager refers to odds on the straight-up outcome of a game with no consideration to apoint 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 the "real odds" or 100% probability of an event occurring. This 100% book is displayed without anybookmaker'sprofit 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 the true chances (as imagined by the bookmaker) that the event will or will not occur, but are the amount that thebookmaker 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 successfulbettor is less than that represented by the true chance of the event occurring. This profit 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 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 the book has anoverround 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 theexpected value of his profit is positive even if everybody bets on 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 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.[14]
Making a profit ingambling involves predicting 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 a bookmaker's brokerage fee ("vig" orvigorish).
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.
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