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The odds in favor of an event or a proposition are 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 day of the week is a Sunday are one to six, which is sometimes written 1 : 6.; see section 1.5 of Gelman et al. (2003).
'Odds' are an expression of relative probabilities. Often 'odds' are quoted as odds against, rather than as odds in favor. For example, the probability that a random day is a Sunday is one-seventh (1/7), hence the odds that a random day is a Sunday are 1 : 6. The odds against a random day being a Sunday are 6 : 1. 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.
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.
Stating "odds against" is a convenient way to propose a bet. When a bookmaker offers betting odds of 6 : 1 against some event occurring, it means that he is prepared to pay out a prize of six times the stake, and return the stake as well, to anyone who places a bet, by making the stake, that the event will occur. If the event does not occur, then the bookmaker keeps the stake. For example, a winning bet of 10 at 6 : 1 against 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 bookmaker's 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.
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 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 first example above, 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.
Example #2: There are 5 red marbles, 2 green marbles, and 8 yellow marbles. What are the odds against picking a yellow marble?
Answer: 7 : 8
Chances versus odds 
Odds of so many to so many on (or against) some event refers to the ratio of numbers of (equal) chances in favor and against (or vice-versa); chances of so many, in so many refers to the number of (equal) chances in favour relative to the number for and against combined. For example, example #1 above, the "chance of picking a blue marble is 2 in 15", the "odds on picking a blue marble are 2 to 13", "the odds against picking a blue marble are 13 to 2". Odds of 1 to 5 corresponds to 1 chance in 6. The words odds and chances are often interchangeably used to vaguely indicate some measure of probability; the intended meaning of the writer has to be deduced by noting whether the preposition between the two numbers is to or in.
Presentation of odds 
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, a probability 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 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 (as imagined by the bookmaker) that the event will or will not occur, but are the amount that the bookmaker will pay out on a winning bet, together with the required stake. For instance, if the bookmaker offers odds of 4:6 against a certain horse winning a race, this means that he'll accept a $6 dollar stake in return for a payoff of $4, plus return of the stake, if the horse wins. If the horse loses, the bookmaker keeps the stake. 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%. 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 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 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).
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.
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 40 units, or a profit of 30 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 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 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).
Knew that we ventured on such dangerous seas
That if we wrought out life 'twas ten to one
See also 
- Galton box
- Gaming mathematics
- Formal mathematical specification of logistic regression
- Mathematics of bookmaking
- Odds algorithm
- Optimal stopping
- Statistical Soccer (Football) Predictions
- Andrew Gelman, John B. Carlin, Hal S. Stern, and Donald B. Rubin (2003), "Bayesian Data Analysis", Second Edition, CRC Press.
- Wolfram MathWorld. "Wolfram MathWorld (Odds)". Wolfram Research Inc. Retrieved 16 May 2012.
- Multi-State Lottery Association. "Welcome to Powerball - Prizes". Multi-State Lottery Association. Retrieved 16 May 2012.
- Lisa Grossman (October 28, 2010). "Odds of Finding Earth-Size Exoplanets Are 1-in-4". Wired. Retrieved 16 May 2012.
- Wolfram Alpha. "Wolfram Alpha (Poker Probabilities)". Wolfram Alpha. Retrieved 16 May 2012.
- James, Franklin (2001). The Science of Conjecture: Evidence and Probability Before Pascal. Baltimore: The Johns Hopkins University Press. pp. 280–281.