Editing Odds (section)

==Statistical usage==
[[File:probability_vs_odds.svg|thumb|right|Calculation of probability (risk) vs odds]]
In statistics, odds are an expression of relative probabilities, generally quoted as the odds ''in favor''. The odds (in favor) of an [[event (probability theory)|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|equally probable outcomes]], this is the ratio of the number of [[Outcome (probability)|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 [[random variable|randomly chosen]] day of the week is during 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.<ref name="Wolfram"/><ref name="Gelman"/> Conversely, given odds as a ratio of integers, this can be represented by a probability space of a finite number of equally probable outcomes. These definitions are equivalent, since dividing both terms in the ratio by the number of outcomes yields the probabilities: <math>2:5 = (2/7):(5/7).</math> Conversely, the odds against is the opposite ratio. For example, the odds against a random day of the week being during 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 probable) outcomes in favor and against (or vice versa); "chances of so many [outcomes], ''in'' so many [outcomes]" refers to ''probability''—the number of (equally probable) 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''.<ref name="Powerball"/><ref name="Wired"/><ref name="Wolfram2"/>

===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 [[Fraction (mathematics)|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 (number of outcomes that are wins:number of outcomes that are losses), the odds in favor (as a number) <math>o_f</math> and odds against (as a number) <math>o_a</math> can be computed by simply dividing, and are [[multiplicative inverse]]s:
:<math>
\begin{align}
o_f &= W/L = 1/o_a\\
o_a &= L/W = 1/o_f\\
o_f \cdot o_a &= 1
\end{align}
</math>

Analogously, given odds as a ratio, the probability of success {{mvar|p}} or failure {{mvar|q}} can be computed by dividing, and the probability of success and probability of failure sum to [[Unity (mathematics)|unity]] (one), as they are the only possible outcomes. In case of a finite number of equally probable outcomes, this can be interpreted as the number of outcomes where the event occurs divided by the total number of events:
:<math>
\begin{align}
p &= W/(W+L) = 1 - q\\
q &= L/(W+L) = 1 - p\\
p + q &= 1
\end{align}
</math>

Given a probability ''p,'' the odds as a ratio is <math>p:q</math> (probability of success to probability of failure), and the odds as numbers can be computed by dividing:
:<math>
\begin{align}
o_f &= p/q = p/(1-p) = (1-q)/q\\
o_a &= q/p = (1-p)/p = q/(1-q)
\end{align}
</math>

Conversely, given the odds as a number <math>o_f,</math> this can be represented as the ratio <math>o_f:1,</math> or conversely <math>1:(1/o_f) = 1:o_a,</math> from which the probability of success or failure can be computed:
:<math>
\begin{align}
p &= o_f/(o_f+1) = 1/(o_a + 1)\\
q &= o_a/(o_a+1) = 1/(o_f + 1)
\end{align}
</math>
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 = 0.0101... = 0.{{overline|01}}), while odds of 1 ''to'' 100 (1/100 = 0.01) is the same as a probability of 1 ''in'' 101 (1/101 = 0.00990099... = 0.{{overline|0099}}). 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:
{| class=wikitable
! odds (ratio) || <math>o_f</math> || <math>o_a</math> || <math>p</math> || <math>q</math>
|-
|1:1 || 1 || 1 || 50% || 50%
|-
|0:1 || 0 || ''∞'' || 0% || 100%
|-
|1:0 || ''∞'' || 0 || 100% || 0%
|-
|2:1 || 2 || 0.5 || 66.{{overline|66}}% || 33.{{overline|33}}%
|-
|1:2 || 0.5 || 2 || 33.{{overline|33}}% || 66.{{overline|66}}%
|-
|4:1 || 4 || 0.25 || 80% || 20%
|-
|1:4 || 0.25 || 4 || 20% || 80%
|-
|colspan=5|
|-
|9:1 || 9 || 0.{{overline|1}} || 90% || 10%
|-
|10:1 || 10 || 0.1 || 90.{{overline|90}}% || 9.{{overline|09}}%
|-
|99:1 || 99 || 0.{{overline|01}} || 99% || 1%
|-
|100:1 || 100 || 0.01 || 99.{{overline|0099}}% || 0.{{overline|9900}}%
|}

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 transformation]]s (fractional linear transformations). They are thus [[Möbius transformation#Specifying a transformation by three points|specified by three points]] ([[Sharply multiply transitive|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 transform]]s. Converting odds to probability fixes 0, sends infinity to 1, and sends 1 to .5 (even odds are 50% probable), and conversely; this is a [[parabolic transform]].

===Applications===
In [[probability theory]] and statistics, odds and similar ratios may be more natural or more convenient than probabilities. 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. This is particularly important in the [[logistic regression|logistic model]], in which the log-odds of the target variable are a [[linear combination]] of the observed variables.

Similar ratios are used elsewhere in statistics; of central importance is the [[likelihood ratio]] in [[likelihoodist statistics]], which is used in [[Bayesian statistics]] as the [[Bayes factor]].

Odds are particularly useful 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]].

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 trial]]s. While they have useful mathematical properties, they can produce counter-[[Intuition (knowledge)|intuitive]] results: an event with an 80% probability of occurring is four times ''more probable'' to happen than an event with a 20% probability, but the ''odds'' are 16 times higher on the less probable event (4–1 ''against'', or 4) than on the more probable one (1–4 ''against'', 4-1 ''in favor'', 4–1 ''on'', or 0.25).

;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 <math>\frac{p}{1-p}</math>. 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.

;Example #2:
In the 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 <math>\frac{1-p}{p}</math>. The odds against Sunday are 6:1 or 6/1&nbsp;=&nbsp;6. It is 6 times as probable that a random day is not a Sunday.