Odds

For the alternative rock band, see Odds (band).

"Odds against" redirects here. For the 1966 documentary film, see The Odds Against.

In probability theory and statistics, the odds in favor of an event or a proposition are the quantity ${\frac {p}{1-p}}$, where p is the probability of the event or proposition. The odds against the same event are ${\frac {1-p}{p}}$.

For example, the odds that a random day of the week (7 days) is a Sunday are:

${\frac {1/7}{1-(1/7)}}={\frac {1/7}{6/7}}={\frac {1}{6}},$

but not 1/7. [to avoid confusion with ordinary fractions, this is often shown as 1:6 and called "one to six"]

The odds against Sunday are 6/1 = 6: it is 6 times more likely that a random day is not a Sunday. These 'odds' are actually relative probabilities. Generally 'odds' are not quoted in this format because of the natural confusion with the chance of an event occurring being expressed fractionally as a probability. E.g., the probability of a random day of the week is a Sunday is 'one-seventh' (1/7). A bookmaker may (for his own purposes) use 'odds' of 'one-sixth', 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). However, even in probability theory, 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.

In some games of chance, this 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.

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 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 that the event will occur, but are the amounts that the bookmaker 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 chances of each of the horses winning based on their relative abilities may be 50%, 40% and 10%. These are the relative probabilities of the horses winning and 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 of winning for each of the three horses is 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 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 amounts bet on each of the respective 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).

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 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).

The logarithm of the odds is the logit 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. Shakespeare wrote:

Knew that we ventured on such dangerous seas
That if we wrought out life 'twas ten to one

— William Shakespeare, Henry IV, Part II, Act I, Scene 1 lines 181–2.

See also

• Gambling
• Gaming mathematics
• Logistic regression#Formal_mathematical_specification
• Mathematics of bookmaking
• Odds algorithm
• Optimal stopping
• Statistical Soccer (Football) Predictions