Odds

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Odds are a numerical expression, always consisting of a pair of numbers, used in both gambling and statistics. In statistics, Odds for reflect the likelihood that a particular event will take place. Odds against reflect the likelihood that a particular event will not take place. The usages of the term among statisticians and probabilists on the one hand, versus in the gambling world on the other hand, are not consistent with each other (with the exception of horse racing).[1][2] Conventionally, gambling odds are expressed in the form "X to Y", where X and Y are numbers, and it is implied that the odds are odds against the event on which the gambler is considering wagering. In both gambling and statistics, the 'odds' are a numerical expression of how likely some possible future event is.

In gambling, odds represent the ratio between the amounts staked by parties to a wager or bet.[3] Thus, odds of 6 to 1 mean the first party (normally a bookmaker) is staking six times the amount that the second party is. Thus, gambling 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.

In statistics, the Odds for an event E are defined as a simple function of the probability of that possible event E. One drawback of expressing the uncertainty of this possible event as Odds for is that to regain the probability requires a calculation. The natural way to interpret Odds for (without calculating anything) is as the ratio of events to non-events in the long run. A simple example is that the (statistical) Odds for rolling six with a fair die (one of a pair of dice) are 1 to 5. This is because, if one rolls the die many times, and keeps a tally of the results, one expects 1 six event for every 5 times the die does not show six. For example, if we roll the fair die 600 times, we would very much expect something in the neighborhood of 100 sixes, and 500 of the other five possible outcomes. That is a ratio of 100 to 500, or simply 1 to 5. To express the (statistical) Odds against, the order of the pair is reversed. Hence the Odds against rolling a six with a fair die are 5 to 1. The probability of rolling a six with a fair die is the single number 1/6 or approximately 16.7%.

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. 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.

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.[4] 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.

The sixteenth century polymath Cardano demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes (which implies that the probability of an event is given by the ratio of favourable outcomes to the total number of possible outcomes [5]).

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

{{#invoke:main|main}} "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

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 (2:5), as days of the week form a sample space of seven outcomes, and the event occurs for two of the outcomes (Saturday and Sunday), and not for the other five.[6][7] Conversely, given odds as a ratio of integers, this can be represented by a probability space of a finite number of equally likely outcomes. These definitions are equivalent, since dividing both terms in the ratio by the number of outcomes yields the probabilities: ${\displaystyle 2:5=(2/7):(5/7).}$ Conversely, the odds against is the opposite ratio. For example, the odds against a random day of the week being 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 (or against) [some event]" refers to odds – the ratio of numbers of (equally likely) outcomes in favor and against (or vice versa); "chances of so many [outcomes], in so many [outcomes]" refers to probability – the number of (equally like) 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 (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.[8][9][10]

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 (even odds). 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. 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 (in favor) as the ratio W:L (Wins:Losses), the odds in favor (as a number) ${\displaystyle o_{f}}$ and odds against (as a number) ${\displaystyle o_{a}}$ can be computed by simply dividing, and a multiplicative inverses:

{\displaystyle {\begin{aligned}o_{f}&=W/L=1/o_{a}\\o_{a}&=L/W=1/o_{f}\\o_{f}\cdot o_{a}&=1\end{aligned}}}

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 by the total number of events:

{\displaystyle {\begin{aligned}p&=W/(W+L)=1-q\\q&=L/(W+L)=1-p\\p+q&=1\end{aligned}}}

Given a probability p, the odds as a ratio is ${\displaystyle p:q}$ (probability of success to probability of failure), and the odds as numbers can be computed by dividing:

{\displaystyle {\begin{aligned}o_{f}&=p/q=p/(1-p)=(1-q)/q\\o_{a}&=q/p=(1-p)/p=q/(1-q)\end{aligned}}}

Conversely, given the odds as a number ${\displaystyle o_{f},}$ this can be represented as the ratio ${\displaystyle o_{f}:1,}$ or conversely ${\displaystyle 1:(1/o_{f})=1:o_{a},}$ from which the probability of success or failure can be computed:

{\displaystyle {\begin{aligned}p&=o_{f}/(o_{f}+1)=1/(o_{a}+1)\\q&=o_{a}/(o_{a}+1)=1/(o_{f}+1)\end{aligned}}}

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 (1/100 = 1%) is the same as odds of 1 to 99 (1/99 = 0.01\ldots = 0.Template:Overline), while odds of 1 to 100 (1/100 = 0.01) is the same as a probability of 1 in 101 (1/101 = 0.9090…% = 0.Template:Overline%). 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 are worked out for some simple odds:

 odds (ratio) ${\displaystyle o_{f}}$ ${\displaystyle o_{a}}$ ${\displaystyle p}$ ${\displaystyle q}$ 1:1 1 1 50% 50% 0:1 0 ∞ 0% 100% 1:0 ∞ 0 100% 0% 2:1 2 .5 67% 33% 1:2 .5 2 33% 67% 4:1 4 .25 80% 20% 1:4 .25 4 20% 80% 9:1 9 0.Template:Overline 90% 10% 10:1 10 0.1 90.Template:Overline% 9.Template:Overline% 99:1 99 0.Template:Overline 99% 1% 100:1 100 0.01 99.Template:Overline% 0.Template:Overline%

These transforms have certain special geometric properties: the conversions between odds for and odds against (resp. probability of success with probability of failure) and between odds and probability are all Möbius transformations (fractional linear transformations). They are thus are 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 .5; these are both order 2, hence circular transforms. Converting odds to probability fixes 0, sends infinity to 1, and sends 1 to .5 (even odds are 50% likely), and conversely; this is a parabolic transform.

Applications

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: 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).

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.

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 ${\displaystyle {\frac {p}{1-p}}}$. 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 ${\displaystyle {\frac {1-p}{p}}}$. 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?