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In probability theory, an event is a subset of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. [1] A single outcome may be an element of many different events, [2] and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes. [3]
The event space could be the set of all subsets of the sample space, which would then contain simple events such as {} ("the die lands on 5"), as well as complex events such as {,,} ("the die lands on an even number"). The probability function would then map each event to the number of outcomes in that event divided by 6 – so for example ...
This event encompasses the possibility of any number except five being rolled. The mutually exclusive event {5} has a probability of 1/6, and the event {1,2,3,4,5,6} has a probability of 1, that is, absolute certainty. When doing calculations using the outcomes of an experiment, it is necessary that all those elementary events have a number ...
While they have useful mathematical properties, they can produce counter-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, or 4–1 on ...
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 ...
In probability theory, the chain rule [1] (also called the general product rule [2] [3]) describes how to calculate the probability of the intersection of, not necessarily independent, events or the joint distribution of random variables respectively, using conditional probabilities.
A tail event is then by definition an event which is measurable with respect to the σ-algebra generated by all X n, but which is independent of any finite number of X n. That is, a tail event is precisely an element of the terminal σ-algebra ⋂ n = 1 ∞ G n {\displaystyle \textstyle {\bigcap _{n=1}^{\infty }G_{n}}} .
The event that contains all possible outcomes of an experiment is its sample space. A single outcome can be a part of many different events. [4] Typically, when the sample space is finite, any subset of the sample space is an event (that is, all elements of the power set of the sample space are defined as events