Editing Odds (section)

===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.