In
probability theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
, odds provide a measure of the probability of a particular outcome. Odds are commonly used in
gambling
Gambling (also known as betting or gaming) is the wagering of something of Value (economics), value ("the stakes") on a Event (probability theory), random event with the intent of winning something else of value, where instances of strategy (ga ...
and
statistics
Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
. For example for an event that is 40% probable, one could say that the odds are or
When
gambling
Gambling (also known as betting or gaming) is the wagering of something of Value (economics), value ("the stakes") on a Event (probability theory), random event with the intent of winning something else of value, where instances of strategy (ga ...
, 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 , pays out , which is called When
Moneyline odds
Fixed-odds betting is a form of gambling where individuals place bets on the outcome of an event, such as sports matches or horse races, at predetermined odds. In fixed-odds betting, the odds are fixed and determined at the time of placing the b ...
are quoted as a positive number , it means that a wager pays When Moneyline odds are quoted as a negative number , it means that a wager pays
Odds have a simple relationship with
probability
Probability is a branch of mathematics and statistics 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 e ...
. When probability is expressed as a number between 0 and 1, the relationships between probability and odds are as follows. Note that if probability is to be expressed as a percentage these probability values should be multiplied by 100%.
*" in " means that the probability is .
*" to in favor" and " to on" mean that the probability is .
*" to against" means that the probability is .
*"pays to " means that the bet is a fair bet if the probability is .
*"pays for " means that the bet is a fair bet if the probability is .
*"pays " (
moneyline odds
Fixed-odds betting is a form of gambling where individuals place bets on the outcome of an event, such as sports matches or horse races, at predetermined odds. In fixed-odds betting, the odds are fixed and determined at the time of placing the b ...
) means that the bet is fair if the probability is .
*"pays " (
moneyline odds
Fixed-odds betting is a form of gambling where individuals place bets on the outcome of an event, such as sports matches or horse races, at predetermined odds. In fixed-odds betting, the odds are fixed and determined at the time of placing the b ...
) means that the bet is fair if the probability is .
The numbers for odds can be scaled. If is any positive number then is the same as and similarly if "to" is replaced with "in" or "for". For example, is the same as both and
When the value of the probability (between 0 and 1; not a percentage) can be written as a fraction then the odds can be said to be or and these can be scaled to equivalent odds. Similarly, fair betting odds can be expressed as or
History
The language of odds, such as the use of phrases like "ten to one" for
intuitive
Intuition is the ability to acquire knowledge without recourse to conscious reasoning or needing an explanation. Different fields use the word "intuition" in very different ways, including but not limited to: direct access to unconscious knowledg ...
ly estimated risks, is found in the sixteenth century, well before the development of
probability theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
.
Shakespeare
William Shakespeare ( 23 April 1564 – 23 April 1616) was an English playwright, poet and actor. He is widely regarded as the greatest writer in the English language and the world's pre-eminent dramatist. He is often called England's natio ...
wrote:
The sixteenth-century
polymath
A polymath or polyhistor is an individual whose knowledge spans many different subjects, known to draw on complex bodies of knowledge to solve specific problems. Polymaths often prefer a specific context in which to explain their knowledge, ...
Cardano 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 the
ratio
In mathematics, a ratio () shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
of favourable outcomes to the total number of possible outcomes.
Statistical usage
In statistics, odds are an expression of relative probabilities, generally quoted as the odds ''in favor''. The odds (in favor) of an
event
Event may refer to:
Gatherings of people
* Ceremony, an event of ritual significance, performed on a special occasion
* Convention (meeting), a gathering of individuals engaged in some common interest
* Event management, the organization of eve ...
or a
proposition
A proposition is a statement that can be either true or false. It is a central concept in the philosophy of language, semantics, logic, and related fields. Propositions are the object s denoted by declarative sentences; for example, "The sky ...
is the ratio of the probability that the event will happen to the probability that the event will not happen. Mathematically, this is a
Bernoulli trial
In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is ...
, as it has exactly two outcomes. In case of a finite
sample space
In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually den ...
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 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 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 prepositions ''to'' and ''in:'' "odds of so many ''to'' so many on (or against)
ome event refers to ''odds''—the ratio of numbers of (equally probable) outcomes in favor and against (or vice versa); "chances of so many
utcomes ''in'' so many
utcomes refers to ''probability''—the number of (equally probable) outcomes in favour relative to 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''.
Mathematical relations
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 (different
fractions
A fraction (from , "broken") represents a part of 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, thre ...
can represent the same
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example,
The set of all ...
). 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 are
multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a ra ...
s:
:
Analogously, given odds as a ratio, the probability of success or failure can be computed by dividing, and 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
(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 1 ''in'' 100 (1/100 = 1%) is the same as odds of 1 ''to'' 99 (1/99 = 0.0101... = 0.), 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.). 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:
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 transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
f(z) = \frac
of one complex number, complex variable ; here the coefficients , , , are complex numbers satisfying .
Geometrically ...
s (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 transform
Circular may refer to:
* The shape of a circle
* ''Circular'' (album), a 2006 album by Spanish singer Vega
* Circular letter (disambiguation), a document addressed to many destinations
** Government circular, a written statement of government pol ...
s. Converting odds to probability fixes 0, sends infinity to 1, and sends 1 to .5 (even odds are 50% probable), and conversely; this is a
parabolic transform
Parabolic usually refers to something in a shape of a parabola, but may also refer to a parable.
Parabolic may refer to:
*In mathematics:
**In elementary mathematics, especially elementary geometry:
**Parabolic coordinates
**Parabolic cylindrical ...
.
Applications
In
probability theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
and statistics, odds and similar ratios may be more natural or more convenient than probabilities. In some cases the
log-odds
In statistics, the logit ( ) function is the quantile function associated with the standard logistic distribution. It has many uses in data analysis and machine learning, especially in data transformations.
Mathematically, the logit is the ...
are used, which is the
logit
In statistics, the logit ( ) function is the quantile function associated with the standard logistic distribution. It has many uses in data analysis and machine learning, especially in Data transformation (statistics), data transformations.
Ma ...
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 the log-odds of the target variable are a
linear combination
In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of the observed variables.
Similar ratios are used elsewhere in statistics; of central importance is the
likelihood ratio in
likelihoodist statistics
Likelihoodist statistics or likelihoodism is an approach to statistics that exclusively or primarily uses the likelihood function. Likelihoodist statistics is a more minor school than the main approaches of Bayesian statistics and frequentist sta ...
, which is used in
Bayesian statistics
Bayesian statistics ( or ) is a theory in the field of statistics based on the Bayesian interpretation of probability, where probability expresses a ''degree of belief'' in an event. The degree of belief may be based on prior knowledge about ...
as the
Bayes factor
The Bayes factor is a ratio of two competing statistical models represented by their evidence, and is used to quantify the support for one model over the other. The models in question can have a common set of parameters, such as a null hypothesis ...
.
Odds are particularly useful in problems of sequential decision making, as for instance in problems of how to stop (online) on a last specific event which is solved by the
odds algorithm
In decision theory, the odds algorithm (or Bruss algorithm) is a mathematical method for computing optimal strategies for a class of problems that belong to the domain of optimal stopping problems. Their solution follows from the ''odds strategy' ...
.
The odds are a
ratio
In mathematics, a ratio () shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
of probabilities; an
odds ratio
An odds ratio (OR) is a statistic that quantifies the strength of the association between two events, A and B. The odds ratio is defined as the ratio of the odds of event A taking place in the presence of B, and the odds of A in the absence of B ...
is a ratio of odds, that is, a ratio of ratios of probabilities. Odds-ratios are often used in analysis of
clinical trial
Clinical trials are prospective biomedical or behavioral research studies on human subject research, human participants designed to answer specific questions about biomedical or behavioral interventions, including new treatments (such as novel v ...
s. While they have useful mathematical properties, they can produce counter-
intuitive
Intuition is the ability to acquire knowledge without recourse to conscious reasoning or needing an explanation. Different fields use the word "intuition" in very different ways, including but not limited to: direct access to unconscious knowledg ...
results: an event with an 80% probability of occurring is four times ''more probable'' to happen than an event with 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 ''against'', 4-1 ''in favor'', 4–1 ''on'', or 0.25).
;Example #1: There are 5 pink marbles, 2 blue marbles, and 8 purple marbles. What are the odds in favor of picking a blue marble?
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
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
and
statistics
Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
, where the variable ''p'' is the
probability
Probability is a branch of mathematics and statistics 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 e ...
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.
;Example #2:
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 as ''p'' 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.
Gambling usage
On a
coin toss
A coin is a small object, usually round and flat, used primarily as a medium of exchange or legal tender. They are standardized in weight, and produced in large quantities at a mint in order to facilitate trade. They are most often issued by a ...
or a
match race
A match race is a race between two competitors, going head-to-head.
In sailboat racing it is differentiated from a fleet race, which almost always involves three or more competitors competing against each other, and team racing where teams cons ...
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 a
bookmaker
A bookmaker, bookie, or turf accountant is an organization or a person that accepts and pays out bets on sporting and other events at agreed-upon odds
In probability theory, odds provide a measure of the probability of a particular outco ...
, and an individual, rather than between individuals. Different traditions have grown up in how to express odds to customers.
Fractional odds
Favoured by
bookmaker
A bookmaker, bookie, or turf accountant is an organization or a person that accepts and pays out bets on sporting and other events at agreed-upon odds
In probability theory, odds provide a measure of the probability of a particular outco ...
s in the
United Kingdom
The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom (UK) or Britain, is a country in Northwestern Europe, off the coast of European mainland, the continental mainland. It comprises England, Scotlan ...
and
Ireland
Ireland (, ; ; Ulster Scots dialect, Ulster-Scots: ) is an island in the North Atlantic Ocean, in Northwestern Europe. Geopolitically, the island is divided between the Republic of Ireland (officially Names of the Irish state, named Irelan ...
, and also common in
horse racing
Horse racing is an equestrian performance activity, typically involving two or more horses ridden by jockeys (or sometimes driven without riders) over a set distance for competition. It is one of the most ancient of all sports, as its bas ...
, 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 (''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'', or ''4 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 as ''evens'' or ''even money''.
The
numerator
A fraction (from , "broken") represents a part of 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, thre ...
and
denominator
A fraction (from , "broken") represents a part of 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, thre ...
of fractional odds are often
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s, 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 the
lowest common denominator
In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the lowest common multiple of the denominators of a set of fractions. It simplifies adding, subtracting, and comparing fractions.
Description
The l ...
. 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 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 the net return.
Decimal odds
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
Europe is a continent located entirely in the Northern Hemisphere and mostly in the Eastern Hemisphere. It is bordered by the Arctic Ocean to the north, the Atlantic Ocean to the west, the Mediterranean Sea to the south, and Asia to the east ...
,
Australia
Australia, officially the Commonwealth of Australia, is a country comprising mainland Australia, the mainland of the Australia (continent), Australian continent, the island of Tasmania and list of islands of Australia, numerous smaller isl ...
,
New Zealand
New Zealand () is an island country in the southwestern Pacific Ocean. It consists of two main landmasses—the North Island () and the South Island ()—and List of islands of New Zealand, over 600 smaller islands. It is the List of isla ...
,
Canada
Canada is a country in North America. Its Provinces and territories of Canada, ten provinces and three territories extend from the Atlantic Ocean to the Pacific Ocean and northward into the Arctic Ocean, making it the world's List of coun ...
, and
Singapore
Singapore, officially the Republic of Singapore, is an island country and city-state in Southeast Asia. The country's territory comprises one main island, 63 satellite islands and islets, and one outlying islet. It is about one degree ...
, 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.
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, 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
A betting exchange is a marketplace for customers to bet on the outcome of discrete events. Betting exchanges offer the same opportunities to bet as a bookmaker with a few differences. Gamblers can buy (also known as "back") and sell (also known ...
because they are the easiest to work with for trading, as they reflect the reciprocal 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
Moneyline odds are favoured by American bookmakers. The figure quoted is either positive or negative.
* When moneyline odds are positive, the figure indicates the net winnings for a $100 wager (this is done for an outcome that is considered less probable to happen than not). For example, net winnings of 4/1 would be quoted as +400.
* When moneyline odds are negative, the figure indicates how much money must be wagered to for a net winning of $100 (this is done for an outcome that is considered more probable to happen than not). For example, net winnings of 1/4 would be quoted as −400.
Moneyline odds are often referred to as ''American odds''. A "moneyline" wager refers to odds on the straight-up outcome of a game with no consideration to a
point spread
Spread betting is any of various types of wagering on 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, such as fixed-odds (or money-line) betting or parimutuel betting.
...
. 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
Wholesale odds are the "real odds" or 100% probability of an event occurring. This 100% book is displayed without any
bookmaker
A bookmaker, bookie, or turf accountant is an organization or a person that accepts and pays out bets on sporting and other events at agreed-upon odds
In probability theory, odds provide a measure of the probability of a particular outco ...
's
profit margin
Profit margin is a financial ratio that measures the percentage of profit earned by a company in relation to its revenue. Expressed as a percentage, it indicates how much profit the company makes for every dollar of revenue generated. Profit margi ...
, often referred to as a bookmaker's "
overround" built in.
A "wholesale odds"
index
Index (: indexes or indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on the Halo Array in the ...
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.
Gambling odds vis-à-vis probabilities
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 the
bookmaker
A bookmaker, bookie, or turf accountant is an organization or a person that accepts and pays out bets on sporting and other events at agreed-upon odds
In probability theory, odds provide a measure of the probability of a particular outco ...
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 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 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
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
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.
Making a profit in
gambling
Gambling (also known as betting or gaming) is the wagering of something of Value (economics), value ("the stakes") on a Event (probability theory), random event with the intent of winning something else of value, where instances of strategy (ga ...
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" or
vigorish
Vigorish (also called the cut, the house edge, juice, the margin, the take, under-juice, or the vig) is the fee charged by a bookmaker for accepting a gambler's wager. In American English, it can also refer to the interest owed a loanshark in con ...
).
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.
See also
*
Odds ratio
An odds ratio (OR) is a statistic that quantifies the strength of the association between two events, A and B. The odds ratio is defined as the ratio of the odds of event A taking place in the presence of B, and the odds of A in the absence of B ...
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Odds algorithm
In decision theory, the odds algorithm (or Bruss algorithm) is a mathematical method for computing optimal strategies for a class of problems that belong to the domain of optimal stopping problems. Their solution follows from the ''odds strategy' ...
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Galton board
The Galton board, also known as the Galton box or quincunx or bean machine (or incorrectly Dalton board), is a device invented by Francis Galton to demonstrate the central limit theorem, in particular that with sufficient sample size the binomi ...
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Gambling mathematics
The mathematics of gambling is a collection of probability applications encountered in Game of chance, games of chance and can be included in game theory. From a mathematical point of view, the games of chance are experiments generating various t ...
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Formal mathematical specification of logistic regression
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Optimal stopping
In mathematics, the theory of optimal stopping or early stopping
: (For French translation, secover storyin the July issue of ''Pour la Science'' (2009).) is concerned with the problem of choosing a time to take a particular action, in order to ...
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Parimutuel betting
Parimutuel betting, or pool betting, is a betting system in which all bets of a particular type are placed together in a pool; taxes and the ''house-take'', or ''vigorish'', are deducted, and payoff odds are calculated by sharing the pool among a ...
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Statistical association football predictions
Statistical association football prediction is a method used in sports betting to predict the outcome of football matches by means of statistical tools. The goal of statistical match prediction is to outperform the predictions of bookmakers, who ...
References
{{Use dmy dates, date=September 2019
Randomness
Statistical ratios
Wagering