Odds

# Odds

Overview
The odds in favor of an event
Event (probability theory)
In probability theory, an event is a set of outcomes to which a probability is assigned. Typically, when the sample space is finite, any subset of the sample space is an event...

or a proposition
Proposition
In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...

are expressed as the ratio
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...

of a pair of integers, which is the ratio of the probability that an event will happen to the probability that it will not happen. For example, the odds that a randomly chosen
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

day of the week is a Sunday are one to six, which is sometimes written 1:6, or 1/6.
In probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, where the variable p is the probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

in favor of the event, and the probability against the event is therefore 1-p, "the odds" of the event are the quotient of the two, or .
Discussion

Encyclopedia
The odds in favor of an event
Event (probability theory)
In probability theory, an event is a set of outcomes to which a probability is assigned. Typically, when the sample space is finite, any subset of the sample space is an event...

or a proposition
Proposition
In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...

are expressed as the ratio
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...

of a pair of integers, which is the ratio of the probability that an event will happen to the probability that it will not happen. For example, the odds that a randomly chosen
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

day of the week is a Sunday are one to six, which is sometimes written 1:6, or 1/6.
In probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, where the variable p is the probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

in favor of the 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 likelihood 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 example just given, 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 likely that a random day is not a Sunday. Hence 'odds' are an expression of relative probabilities. Generally 'odds' are quoted in this format (odds against) rather than as odds in favor of, because of the possibility of confusion of the latter with the fractional probability of an event occurring. E.g., the probability of a random day of the week is a Sunday is 'one-seventh' (1/7). A bookmaker
Bookmaker
A bookmaker, or bookie, is an organization or a person that takes bets on sporting and other events at agreed upon odds.- Range of events :...

may (for his own purposes) use 'odds' of 'one-sixth', but the overwhelming everyday use by most people is odds of the form 6 to 1, 6-1, 6:1, or 6/1 (all read as 'six-to-one') where the first figure represents the number of ways of failing to achieve the outcome and the second figure is the number of ways of achieving a favorable outcome: thus these are "odds against". In other words, an event with m to n "odds against" would have probability n/(m + n), while an event with m to n "odds on" would have probability m/(m + n).
Even in probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

, odds may be more natural or more convenient than probabilities. This is in particular the case 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
Odds algorithm
The odds-algorithm is a mathematical method for computing optimalstrategies for a class of problems that belong to the domain of optimal stopping problems. Their solution follows from the odds-strategy, and the importance of the...

.

In some games of chance, using odds against is also the most convenient way to understand what winnings will be paid if the selection is successful: the winner will be paid 'six' of whatever stake unit was bet for each 'one' of the stake unit wagered. For example, a winning bet of 10 at 6/1 will win '6 × 10 = 60' with the original 10 stake also being returned. Betting odds are skewed to ensure that the bookmaker makes a profit—if true odds were offered the bookmaker would break even in the long run—so the numbers do not represent the true odds.

"Odds on" means that the event is more likely to happen than not. This is sometimes expressed with the smaller number first (1:2) but more often using the word "on" (2:1 on) meaning that the event is twice as likely to happen as not.

### Decimal presentation

Taking an event with a 1 in 5 probability of occurring (i.e. a probability of 1/5, 0.2 or 20%), then the odds are 0.2 / (1 − 0.2) = 0.2 / 0.8 = 0.25. This figure (0.25) represents the monetary stake necessary for a person to gain one (monetary) unit on a successful wager when offered fair odds. This may be scaled up by any convenient factor to give whole number values. For example, if a stake of 0.25 wins 1 unit, then scaling by a factor of four means a stake of 1 wins 4 units.

### Ratio presentation

Fixed odds gambling tends to represent the probability as fractional odds, and excludes the stake. For example 0.20 is represented as "4 to 1 against" (written as 4-1, 4:1, or 4/1), since there are five outcomes of which four are unsuccessful. Thus the stake returned must be added to the odds to compute the entire return of a successful bet. In craps
Craps
Craps is a dice game in which players place wagers on the outcome of the roll, or a series of rolls, of a pair of dice. Players may wager money against each other or a bank...

the payout would be represented as "5 for 1", and in moneyline odds as +400 representing the gain from a 100 stake.

By contrast, for an event with a 4 in 5 probability of occurring (i.e. a probability of 4/5, 0.8 or 80%), then the odds are 0.8 / (1 − 0.8) = 4. If one bets 4 units at these odds and the event occurs, one receives back 1 unit plus the original unit 4 units stake. This would be presented in fractional odds of "4 to 1 on' (written as 1/4 or 1–4), in decimal odds as 1.25 to include the returned stake, in craps as "5 for 4", and in moneyline odds as −400 representing the stake necessary to gain 100.

Fixed odds are not necessarily presented in the lowest possible terms; if there is a pattern of odds of 5–4, 7–4 and so on, odds which are mathematically 3–2 are more easily compared if expressed in the mathematically equivalent form 6–4. Similarly, 10–3 may be stated as 100–30.

## Gambling odds versus probabilities

In gambling, the odds on display do not represent the true chances that the event will occur, but are the amounts that the bookmaker
Bookmaker
A bookmaker, or bookie, is an organization or a person that takes bets on sporting and other events at agreed upon odds.- Range of events :...

will pay out on winning bets. In formulating his 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 'over-round' 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%. These are simply the bookmaker's 'odds' multiplied by 100% for convenience. 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 by the bookmaker he may decide to increase the values to 60%, 50% and 20% for the three horses, representing odds against of 4-6, 1-1 and 4-1. These values now total 130%, meaning that the book has an overround
Mathematics of bookmaking
In betting parlance, making a book is the practice of laying bets on the various possible outcomes of a single event. The term originates from the practice of recording such wagers in a hard-bound ledger and gives the English language the term bookmaker for the person laying the bets and thus...

of 30 (130 − 100). This value of 30 represents the amount of profit for the bookmaker if he accepts bets in the correct proportions on each of the horses. The art of bookmaking is that he will take in, for example, \$130 in wagers and only pay \$100 back (including stakes) no matter which horse wins.

Profiting in gambling
Gambling
Gambling is the wagering of money or something of material value on an event with an uncertain outcome with the primary intent of winning additional money and/or material goods...

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
Vigorish, or simply the vig, also known as juice or the take, is the amount charged by a bookmaker, or bookie, for his services. In the United States it also means the interest on a shark's loan. The term is Yiddish slang originating from the Russian word for winnings, выигрыш vyigrysh...

).

## Even odds

The terms "even odds", "even money" or simply "evens" (1 to 1, or 2 for 1) imply that the payout will be one unit per unit wagered plus the original stake, that is, 'double-your-money'. Assuming there is no bookmaker fee or built-in profit margin, the actual probability of winning is 50%. The term "better than even odds" (or "better than evens") looks at it from the perspective of a gambler rather than a statistician. If the odds are Evens (1–1), and one bets 10 units, one would be returned 20 units, profiting 10 units. If the gamble was paying 4-1 and the event occurred, one would make a profit of 40 units. So, it is "better than evens" from the gambler's perspective because it pays out more than one-for-one. If an event is more likely to occur than an even chance, then the odds will be "worse than evens", and the bookmaker will pay out less than one-for-one.

In popular parlance surrounding uncertain events, the expression "better than evens" usually implies a better than (greater than) 50% chance of the event occurring, which is exactly the opposite of the meaning of the expression when used in a gaming context.

The odds are a ratio
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...

of probabilities; an odds ratio
Odds ratio
The odds ratio is a measure of effect size, describing the strength of association or non-independence between two binary data values. It is used as a descriptive statistic, and plays an important role in logistic regression...

is a ratio of odds, that is, a ratio of ratios of probabilities. Odds-ratios are often used in analysis of clinical trial
Clinical trial
Clinical trials are a set of procedures in medical research and drug development that are conducted to allow safety and efficacy data to be collected for health interventions...

s. While they have useful mathematical properties, they can produce counter-intuitive
Intuition (knowledge)
Intuition is the ability to acquire knowledge without inference or the use of reason. "The word 'intuition' comes from the Latin word 'intueri', which is often roughly translated as meaning 'to look inside'’ or 'to contemplate'." Intuition provides us with beliefs that we cannot necessarily justify...

results: an event with an 80% probability of occurring is four times more likely to happen than an event with a 20% probability, but the odds are 16 times higher on the less likely event (4–1 against, or 4) than on the more likely one (1–4, or 4–1 on, or 0.25).

The logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

of the odds is the logit
Logit
The logit function is the inverse of the sigmoidal "logistic" function used in mathematics, especially in statistics.Log-odds and logit are synonyms.-Definition:The logit of a number p between 0 and 1 is given by the formula:...

of the probability.

## Historical

The language of odds such as "ten to one" for intuitively estimated risks is found in the sixteenth century, well before the invention of mathematical probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

. Shakespeare wrote:

• Galton box
• Gambling
Gambling
Gambling is the wagering of money or something of material value on an event with an uncertain outcome with the primary intent of winning additional money and/or material goods...

• Gaming mathematics
Gaming mathematics
Gaming mathematics, also referred to as the mathematics of gambling, is a collection of probability applications encountered in games of chance and can be included in applied mathematics...

• Formal mathematical specification of logistic regression
• Mathematics of bookmaking
Mathematics of bookmaking
In betting parlance, making a book is the practice of laying bets on the various possible outcomes of a single event. The term originates from the practice of recording such wagers in a hard-bound ledger and gives the English language the term bookmaker for the person laying the bets and thus...

• Odds algorithm
Odds algorithm
The odds-algorithm is a mathematical method for computing optimalstrategies for a class of problems that belong to the domain of optimal stopping problems. Their solution follows from the odds-strategy, and the importance of the...

• Optimal stopping
Optimal stopping
In mathematics, the theory of optimal stopping is concerned with the problem of choosing a time to take a particular action, in order to maximise an expected reward or minimise an expected cost. Optimal stopping problems can be found in areas of statistics, economics, and mathematical finance...

• Statistical Soccer (Football) Predictions
Statistical Soccer (Football) Predictions
Statistical Football prediction is a method used in sports betting, to predict the outcome of Association football matches by means of statistical tools. The goal of statistical match prediction is to outperform the predictions of bookmakers, who use them to set odds on the outcome of football...

• Betgenius
Betgenius
Betgenius is a software company that provides real-time odds comparison services to the public and bookmakers, they also offer real time data embedded into online advertisements via their Connextra platform. It provides the technology behind the MSN UK gambling channel...