# Odds

(Redirected from Betting odds)
"Odds against" redirects here. For the 1966 documentary film, see The Odds Against.

Odds are a numerical expression used in gambling and statistics to reflect the likelihood that a particular event will take place. Conventionally, they are expressed in the form "X to Y", where X and Y are numbers.

In gambling, odds represent the ratio between the amounts staked by parties to a wager or bet.[1] Thus, odds of 6 to 1 mean the first party (normally a bookmaker) is staking six times the amount that the second party is. In statistics, odds represent the probability that an event will take place. Thus, odds of 6 to 1 mean that there are six possible outcomes in which the event will not take place to every one where it will. In other words, the probability that X will not happen is six times the probability that it will.

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. 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.

## History

The language of odds such as "ten to one" for intuitively estimated risks is found in the sixteenth century, well before the discovery of mathematical probability.[2] Shakespeare wrote:

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

William ShakespeareHenry IV, Part II, Act I, Scene 1 lines 181–2.

## Terminology

Odds are expressed in the form X to Y, where X and Y are numbers. 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. Thus, 6/1, 6-1 and 6:1 are all interchangeable.

### Odds against

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, for example, 6 to 1 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 he himself has staked.

### Odds on

"Odds on" is the opposite of "odds against". 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. For example, if he bets £2, he will be given £1 plus his returned stake of £2, leaving him £1 in profit.

### Even odds

Main article: Even money

"Even odds" occur when the probability of an event happening is exactly the same as it not happening. In common parlance, this is a "50:50 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). "Evens" implies that the payout will be one unit per unit wagered plus the original stake, that is, "double-your-money".

#### Better than/worse than evens

The term "better than evens" (or "worse than evens") varies in meaning depending on context. Looked at from the perspective of a gambler rather than a statistician, "better than evens" means "odds against". If the odds are evens (1/1), and one bets 10 units, one would be returned 20 units, making a profit of 10 units. If the gamble was paying 4/1 and the event occurred, one would make 50 units, or 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.

However, 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.

## Statistical usage

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. 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;[3][4]

'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.

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.[5][6][7]

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).

### Examples

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 $\frac{p}{1-p}$. 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 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 the event with probability given as p are $\frac{1-p}{p}$. 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?

## Gambling usage

The use of odds in gambling arose to facilitate 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 more scope.[citation needed]

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.

### Fractional odds

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 win, relative to his stake.[8] Odds of 4/1 ("four-to-one" or less commonly "four-to-one against") would imply that the bettor stands to make a £400 profit on a £100 stake. If the odds are 1/4 (read "one-to-four", or "four-to-one on"), the bettor will make £25 on a £100 stake. In either case, against or on, should he win, the bettor always receives his original stake back, so if the odds are 4/1 the bettor receives a total of £500 (£400 plus the original £100). Odds of 1/1 are known as evens or even money.

The enumerator and denominator of fractional odds are always integers, thus if the bookmaker's payout was to be £1.25 for every £1 stake, this would be equivalent to £5 for every £4 staked, and the odds would therefore be expressed as 5/4. However, not all fractional odds are traditionally read using the lowest common denominator. For example, given that there is a pattern of odds of 5/4, 7/4, 9/4 and so on, odds which are mathematically 3/2 are more easily compared if expressed in the equivalent form 6/4. Perhaps most unusual is that odds of 10/3 are read as "one-hundred-to-thirty", because "ten-to-three" could be confused with a race time.

Fractional odds are also known as British odds, UK odds,[9] or, in that country, traditional odds. They are typically represented with a "/" but can also be represented with a "-", e.g. 4/1 or 4-1.

A variation of fractional odds is used in craps. Here odds are expressed with the stake included. Thus fractional odds of "four to one against" in craps are represented as "five for one", and odds of "four to one on" as "five for four".[citation needed]

### Decimal odds

Favoured in continental Europe, Australia, New Zealand and Canada, decimal odds differ from fractional odds in that the bettor must first part with their stake in order to make a bet, the figure quoted is the winning amount that would be paid out to the bettor.[10] Therefore, the decimal odds of an outcome are equivalent to the decimal value of the fractional odds plus one.[11] Thus even odds 1/1 are quoted in decimal odds as 2. The 4/1 fractional odds discussed above are quoted as 5, while the 1/4 odds are quoted as 1.25. This is considered to be ideal for parlay betting, because the odds to be paid out are simply the product of the odds for each outcome wagered on. Decimal odds are also favoured by betting exchanges because they are the easiest to work with for trading.

Decimal odds are also known as European odds, digital odds or continental odds.[9]

### Moneyline odds

Moneyline odds are favoured by American bookmakers. There are two possibilities, the figure quote can be either positive or negative.

Positive figures
If the figure quoted is positive, the odds are quoting how much money will be won on a $100 wager (this is done if the odds are better than even). Fractional odds of 4/1 would be quoted as +400, while fractional odds of 1/4 cannot be quoted as a positive figure. Negative figures If the figure quoted is negative, then the moneyline odds are quoting how much money must be wagered to win$100 (this is done if the odds are worse than even). Fractional odds of 1/4 would be quoted as -400, while fractional odds of 4/1 cannot be quoted as a negative figure.

Moneyline odds are often referred to as American odds. Moneyline refers to odds on the straight-up outcome of a game with no consideration to a point spread.

## 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 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.

The logarithm of the odds is the logit of the probability.

## References

1. ^ "Odds, definition of". Oxford Dictionaries. Retrieved 1 May 2014.
2. ^ James, Franklin (2001). The Science of Conjecture: Evidence and Probability Before Pascal. Baltimore: The Johns Hopkins University Press. pp. 280–281.
3. ^ Wolfram MathWorld. "Wolfram MathWorld (Odds)". Wolfram Research Inc. Retrieved 16 May 2012.
4. ^ Gelman, Andrew; Carlin, John B.; Stern, Hal S.; Rubin, Donald B. (2003). "1.5". Bayesian Data Analysis (2nd ed.). CRC Press.
5. ^ Multi-State Lottery Association. "Welcome to Powerball - Prizes". Multi-State Lottery Association. Retrieved 16 May 2012.
6. ^ Lisa Grossman (October 28, 2010). "Odds of Finding Earth-Size Exoplanets Are 1-in-4". Wired. Retrieved 16 May 2012.
7. ^ Wolfram Alpha. "Wolfram Alpha (Poker Probabilities)". Wolfram Alpha. Retrieved 16 May 2012.
8. ^ "Betting School: Understanding Fractional & Decimal Betting Odds". Goal. 10 January 2011. Retrieved 27 March 2014.
9. ^ a b "Betting Odds Format". World Bet Exchange. Retrieved 27 March 2014.
10. ^ D., Chris. "What is Fixed odds betting and Due Column betting?". TBR. Retrieved 27 March 2014.
11. ^ "Fractional Odds". http://betstarter.com/. Retrieved 27 March 2014.