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In probability theory, odds provide a measure of the probability of a particular outcome. Odds are commonly used in gambling and statistics.For example for an event that is 40% probable, one could say that the odds are "2 in 5", "2 to 3 in favor", or "3 to 2 against".
Probability is the branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur. [note 1] [1] [2] This number is often expressed as a percentage (%), ranging from 0% to ...
An odds ratio (OR) is a statistic that quantifies the strength of the association between two events, A and B. The odds ratio is defined as the ratio of the odds of event A taking place in the presence of B, and the odds of A in the absence of B. Due to symmetry, odds ratio reciprocally calculates the ratio of the odds of B occurring in the presence of A, and the odds of B in the absence of A.
Now, if the prevalence of this disease is 9.09%, and if we take that as the prior probability, then the prior odds is about 1:10. So after receiving a positive test result, the posterior odds of having the disease becomes 1:1, which means that the posterior probability of having the disease is 50%.
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms .
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes.Two events are independent, statistically independent, or stochastically independent [1] if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds.
This follows from the definition of independence in probability: the probabilities of two independent events happening, given a model, is the product of the probabilities. This is particularly important when the events are from independent and identically distributed random variables , such as independent observations or sampling with replacement .
The odds algorithm applies to a class of problems called last-success problems. Formally, the objective in these problems is to maximize the probability of identifying in a sequence of sequentially observed independent events the last event satisfying a specific criterion (a "specific event").