Odds


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 or
In gambling, odds are often given as the ratio of the possible net profit to the possible net loss. However in many situations, the possible loss is paid up front and, if the gambler wins, the net win plus the stake is returned. So wagering 2 at, pays out, which is called When Moneyline odds 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. When probability is expressed as a number between 0 and 1, the relationships between probability and odds are as follows. Multiplying these probabilities by 100% gives the percentage.
  • " 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 " means that the bet is fair if the probability is.
  • "pays " 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 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 intuitively estimated risks, is found in the sixteenth century, well before the development of probability theory. Shakespeare wrote:
The sixteenth-century polymath 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 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 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 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 or S and F. For example, the odds that a randomly chosen day of the week is during a weekend are two to five, as days of the week form a sample space of seven outcomes, and the event occurs for two of the outcomes, 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 " refers to odds—the ratio of numbers of outcomes in favor and against ; "chances of so many , in so many " refers to probability—the number of 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 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. Odds can also be expressed as a number, by dividing the terms in the ratio—in this case it is unique. Odds as a ratio, odds as a number, and probability 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 as the ratio W:L, the odds in favor and odds against can be computed by simply dividing, and are multiplicative inverses:
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, 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 , 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 is the same as odds of 1 to 99, while odds of 1 to 100 is the same as a probability of 1 in 101. This is a minor difference if the probability is small, but is a major difference if the probability is large.
These are worked out for some simple odds:
These transforms have certain special geometric properties: the conversions between odds for and odds against and between odds and probability are all Möbius transformations. They are thus specified by three points. 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, and conversely; this is a parabolic transform.

Applications

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 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 on 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 a 20% probability, but the odds are 16 times higher on the less probable event than on the more probable one.
;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 and statistics, where 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. That value may be regarded as the relative probability the event will happen, expressed as a fraction, or a multiple 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 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 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, and an individual, rather than between individuals. Different traditions have grown up in how to express odds to customers.