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This is because, if one rolls the die many times, and keeps a tally of the results, one expects 1 three event for every 5 times the die does not show three i.
For example, if we roll the fair die times, we would very much expect something in the neighborhood of threes, and of the other five possible outcomes.
That is a ratio of to , or simply 1 to 5. To express the statistical odds against, the order of the pair is reversed. Hence the odds against rolling a three with a fair die are 5 to 1.
The gambling and statistical uses of odds are closely interlinked. If a bet is a fair one, then the odds offered to the gamblers will perfectly reflect relative probabilities.
The profit and the expense exactly offset one another and so there is no advantage to gambling over the long run.
If the odds being offered to the gamblers do not correspond to probability in this way then one of the parties to the bet has an advantage over the other.
Casinos , for example, offer odds that place themselves at an advantage, which is how they guarantee themselves a profit and survive as businesses.
The fairness of a particular gamble is more clear in a game involving relatively pure chance, such as the ping-pong ball method used in state lotteries in the United States.
It is much harder to judge the fairness of the odds offered in a wager on a sporting event such as a football match. 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.
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.
Usually, the word "to" is replaced by a symbol for ease of use. This is conventionally either a slash or hyphen , although a colon is sometimes seen.
When the probability that the event will not happen is greater than the probability that it will, then the odds are "against" that event happening.
Odds of 6 to 1, for example, are therefore sometimes said to be "6 to 1 against ". To a gambler, "odds against" means that the amount he or she will win is greater than the amount staked.
It means that the event is more likely to happen than not. This is sometimes expressed with the smaller number first 1 to 2 but more often using the word "on" "2 to 1 on " , meaning that the event is twice as likely to happen as not.
Note that the gambler who bets at "odds on" and wins will still be in profit, as his stake will be returned. In common parlance, this is a " chance".
Guessing heads or tails on a coin toss is the classic example of an event that has even odds. In gambling, it is commonly referred to as " even money " or simply "evens" 1 to 1, or 2 for 1.
The meaning of the term "better than evens" or "worse than evens" depends on context. From the perspective of a gambler rather than a statistician , "better than evens" means "odds against".
If the odds are evens 1: A successful gamble paying out at 4: So this wager 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.
However, in popular parlance surrounding uncertain events, the expression "better than evens" usually implies a greater than percent chance of an event occurring, which is exactly the opposite of the meaning of the expression when used in a gaming context.
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 is a Bernoulli trial , as it has exactly two outcomes. In case of a finite sample space of equally likely 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 a weekend are two to five 2: These definitions are equivalent, since dividing both terms in the ratio by the number of outcomes yields the probabilities: For example, the odds against a random day of the week being a weekend are 5: Odds and probability can be expressed in prose via the prepositions to and in: 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 a ratio of two numbers, in which case it is not unique — scaling both terms by the same factor does not change the proportions: 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 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.
Given odds in favor as the ratio W: 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 likely 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: Thus if expressed as a fraction with a numerator of 1, probability and odds differ by exactly 1 in the denominator: 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 transforms have certain special geometric properties: 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.
Converting odds to probability fixes 0, sends infinity to 1, and sends 1 to. In probability theory and Bayesian statistics , odds may sometimes be more natural or more convenient than probabilities.
This is often 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.
Similar ratios are used elsewhere in Bayesian statistics, such as the Bayes factor. 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: 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.
The odds in favour of a blue marble are 2: One can equivalently say, that the odds are There are 2 out of 15 chances in favour of blue, 13 out of 15 against blue.
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 very 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 Sunday are 6: It is 6 times as likely that a random day is not a Sunday. The use of odds in gambling facilitates betting on events where the relative probabilities of outcomes varied.
For example, 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 sides, betting "at odds" provides a perspective on the relative likelihoods of the possible outcomes.
In the modern era, most fixed odds 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, older eras came with betting odds between people, today which is illegal in most countries, it was referred as "odding", an underground slang word with origins based in the Bronx.
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 he or she win, relative to the stake.
However, not all fractional odds are traditionally read using the lowest common denominator. Fractional odds are also known as British odds, UK odds,  or, in that country, traditional odds.
Odds with a denominator of 1 are often presented in listings as the numerator only.