 In probability theory, an outcome is a possible result of an experiment or trial.[1] Each possible outcome of a particular experiment is unique, and different outcomes are mutually exclusive (only one outcome will occur on each trial of the experiment). All of the possible outcomes of an experiment form the elements of a sample space.[2] For the experiment where we flip a coin twice, the four possible outcomes that make up our sample space are (H, T), (T, H), (T, T) and (H, H), where "H" represents a "heads", and "T" represents a "tails". Outcomes should not be confused with events, which are sets (or informally, "groups") of outcomes. For comparison, we could define an event to occur when "at least one 'heads'" is flipped in the experiment - that is, when the outcome contains at least one 'heads'. This event would contain all outcomes in the sample space except the element (T, T).
Since individual outcomes may be of little practical interest, or because there may be prohibitively (even infinitely) many of them, outcomes are grouped into sets of outcomes that satisfy some condition, which are called "events." The collection of all such events is a sigma-algebra.[3] An event containing exactly one outcome is called an elementary event. 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). However, this approach does not work well in cases where the sample space is uncountably infinite (most notably when the outcome must be some real number). So, when defining a probability space it is possible, and often necessary, to exclude certain subsets of the sample space from being events. Outcomes may occur with probabilities that are between zero and one (inclusively). In a discrete probability distribution whose sample space is finite, each outcome is assigned a particular probability. In contrast, in a continuous distribution, individual outcomes all have zero probability, and non-zero probabilities can only be assigned to ranges of outcomes. Some "mixed" distributions contain both stretches of continuous outcomes and some discrete outcomes; the discrete outcomes in such distributions can be called atoms and can have non-zero probabilities.[5] Under the measure-theoretic definition of a probability space, the probability of an outcome need not even be defined. In particular, the set of events on which probability is defined may be some σ-algebra on S {\displaystyle S} and not necessarily the full power set. Flipping a coin leads to two outcomes that are almost equally likely. Up or down? Flipping a brass tack leads to two outcomes that are not equally likely. In some sample spaces, it is reasonable to estimate or assume that all outcomes in the space are equally likely (that they occur with equal probability). For example, when tossing an ordinary coin, one typically assumes that the outcomes "head" and "tail" are equally likely to occur. An implicit assumption that all outcomes are equally likely underpins most randomization tools used in common games of chance (e.g. rolling dice, shuffling cards, spinning tops or wheels, drawing lots, etc.). Of course, players in such games can try to cheat by subtly introducing systematic deviations from equal likelihood (for example, with marked cards, loaded or shaved dice, and other methods). Some treatments of probability assume that the various outcomes of an experiment are always defined so as to be equally likely.[6] However, there are experiments that are not easily described by a set of equally likely outcomes— for example, if one were to toss a thumb tack many times and observe whether it landed with its point upward or downward, there is no symmetry to suggest that the two outcomes should be equally likely.
In probability theory, an experiment or trial (see below) is any procedure that can be infinitely repeated and has a well-defined set of possible outcomes, known as the sample space.[1] An experiment is said to be random if it has more than one possible outcome, and deterministic if it has only one. A random experiment that has exactly two (mutually exclusive) possible outcomes is known as a Bernoulli trial.[2] When an experiment is conducted, one (and only one) outcome results— although this outcome may be included in any number of events, all of which would be said to have occurred on that trial. After conducting many trials of the same experiment and pooling the results, an experimenter can begin to assess the empirical probabilities of the various outcomes and events that can occur in the experiment and apply the methods of statistical analysis.
Random experiments are often conducted repeatedly, so that the collective results may be subjected to statistical analysis. A fixed number of repetitions of the same experiment can be thought of as a composed experiment, in which case the individual repetitions are called trials. For example, if one were to toss the same coin one hundred times and record each result, each toss would be considered a trial within the experiment composed of all hundred tosses.[3] A random experiment is described or modeled by a mathematical construct known as a probability space. A probability space is constructed and defined with a specific kind of experiment or trial in mind. A mathematical description of an experiment consists of three parts:
An outcome is the result of a single execution of the model. Since individual outcomes might be of little practical use, more complicated events are used to characterize groups of outcomes. The collection of all such events is a sigma-algebra F {\displaystyle \scriptstyle {\mathcal {F}}} . Finally, there is a need to specify each event's likelihood of happening; this is done using the probability measure function, P. Once an experiment is designed and established, ω, from the sample space Ω. All the events in F {\displaystyle \scriptstyle {\mathcal {F}}} that contain the selected outcome ω (recall that each event is a subset of Ω) are said to “have occurred”. The probability function P is defined in such a way that, if the experiment were to be repeated an infinite number of times, the relative frequencies of occurrence of each of the events would approach agreement with the values P assigns them. As a simple experiment, we may flip a coin twice. The sample space (where the order of the two flips is relevant) is {(H, T), (T, H), (T, T), (H, H)} where "H" means "heads" and "T" means "tails". Note that each of (H, T), (T, H), ... are possible outcomes of the experiment. We may define an event which occurs when a "heads" occurs in either of the two flips. This event contains all of the outcomes except (T, T).
|
What is a process that has a number of possible outcomes by which an observation is obtained?
Pos Terkait
Copyright © 2024 ketajaman Inc.
