Outcome (probability)

Last updated November 30, 2024

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).

Sets of outcomes: events

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.

Probability of an outcome

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 $S$ and not necessarily the full power set.

Equally likely outcomes

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.

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In information theory, the entropy of a random variable quantifies the average level of uncertainty or information associated with the variable's potential states or possible outcomes. This measures the expected amount of information needed to describe the state of the variable, considering the distribution of probabilities across all potential states. Given a discrete random variable $, which takes values in the set and is distributed according to, the entropy is where denotes the sum over the variable's possible values. The choice of base for, the logarithm, varies for different applications. Base 2 gives the unit of bits, while base e gives "natural units" nat, and base 10 gives units of "dits", "bans", or "hartleys". An equivalent definition of entropy is the expected value of the self-information of a variable.$

Probability is the branch of mathematics 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. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2.

In probability theory, the sample space of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually denoted using set notation, and the possible ordered outcomes, or sample points, are listed as elements in the set. It is common to refer to a sample space by the labels S, Ω, or U. The elements of a sample space may be numbers, words, letters, or symbols. They can also be finite, countably infinite, or uncountably infinite.

In probability theory, an elementary event, also called an atomic event or sample point, is an event which contains only a single outcome in the sample space. Using set theory terminology, an elementary event is a singleton. Elementary events and their corresponding outcomes are often written interchangeably for simplicity, as such an event corresponding to precisely one outcome.

In probability theory, an event is a set of outcomes of an experiment to which a probability is assigned. A single outcome may be an element of many different events, and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes. An event consisting of only a single outcome is called an elementary event or an atomic event; that is, it is a singleton set. An event that has more than one possible outcome is called a compound event. An event $is said to occur if contains the outcome of the experiment. The probability that an event occurs is the probability that contains the outcome of an experiment. An event defines a complementary event, namely the complementary set, and together these define a Bernoulli trial: did the event occur or not?$

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In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events.

A random variable is a mathematical formalization of a quantity or object which depends on random events. The term 'random variable' in its mathematical definition refers to neither randomness nor variability but instead is a mathematical function in which

In probability theory, a probability space or a probability triple $is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models the throwing of a die.$

In probability and statistics, a Bernoulli process is a finite or infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes only two values, canonically 0 and 1. The component Bernoulli variablesX_i are identically distributed and independent. Prosaically, a Bernoulli process is a repeated coin flipping, possibly with an unfair coin. Every variable X_i in the sequence is associated with a Bernoulli trial or experiment. They all have the same Bernoulli distribution. Much of what can be said about the Bernoulli process can also be generalized to more than two outcomes ; this generalization is known as the Bernoulli scheme.

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In probability theory, an event is said to happen almost surely if it happens with probability 1. In other words, the set of outcomes on which the event does not occur has probability 0, even though the set might not be empty. The concept is analogous to the concept of "almost everywhere" in measure theory. In probability experiments on a finite sample space with a non-zero probability for each outcome, there is no difference between almost surely and surely ; however, this distinction becomes important when the sample space is an infinite set, because an infinite set can have non-empty subsets of probability 0.

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Probability is a measure of the likeliness that an event will occur. Probability is used to quantify an attitude of mind towards some proposition whose truth is not certain. The proposition of interest is usually of the form "A specific event will occur." The attitude of mind is of the form "How certain is it that the event will occur?" The certainty that is adopted can be described in terms of a numerical measure, and this number, between 0 and 1 is called the probability. Probability theory is used extensively in statistics, mathematics, science and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems.

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This glossary of statistics and probability is a list of definitions of terms and concepts used in the mathematical sciences of statistics and probability, their sub-disciplines, and related fields. For additional related terms, see Glossary of mathematics and Glossary of experimental design.

In probability theory, random element is a generalization of the concept of random variable to more complicated spaces than the simple real line. The concept was introduced by Maurice Fréchet who commented that the “development of probability theory and expansion of area of its applications have led to necessity to pass from schemes where (random) outcomes of experiments can be described by number or a finite set of numbers, to schemes where outcomes of experiments represent, for example, vectors, functions, processes, fields, series, transformations, and also sets or collections of sets.”

In common usage, randomness is the apparent or actual lack of definite pattern or predictability in information. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual random events are, by definition, unpredictable, but if there is a known probability distribution, the frequency of different outcomes over repeated events is predictable. For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will tend to occur twice as often as 4. In this view, randomness is not haphazardness; it is a measure of uncertainty of an outcome. Randomness applies to concepts of chance, probability, and information entropy.

In probability theory, an experiment or trial is any procedure that can be infinitely repeated and has a well-defined set of possible outcomes, known as the sample space. 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 possible outcomes is known as a Bernoulli trial.

References

↑ "Outcome - Probability - Math Dictionary". HighPointsLearning. Retrieved 25 June 2013.
↑ Albert, Jim (21 January 1998). "Listing All Possible Outcomes (The Sample Space)". Bowling Green State University. Archived from the original on 16 October 2000. Retrieved June 25, 2013.
↑ Leon-Garcia, Alberto (2008). Probability, Statistics and Random Processes for Electrical Engineering. Upper Saddle River, NJ: Pearson. ISBN 9780131471221.
↑ Pfeiffer, Paul E. (1978). Concepts of probability theory. Dover Publications. p. 18. ISBN 978-0-486-63677-1.
↑ Kallenberg, Olav (2002). Foundations of Modern Probability (2nd ed.). New York: Springer. p. 9. ISBN 0-387-94957-7.
↑ Foerster, Paul A. (2006). Algebra and Trigonometry: Functions and Applications, Teacher's Edition (Classics ed.). Upper Saddle River, NJ: Prentice Hall. p. 633. ISBN 0-13-165711-9.

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[1] "Outcome - Probability - Math Dictionary". HighPointsLearning. Retrieved 25 June 2013.

[2] Albert, Jim (21 January 1998). "Listing All Possible Outcomes (The Sample Space)". Bowling Green State University. Archived from the original on 16 October 2000. Retrieved June 25, 2013.

[3] Leon-Garcia, Alberto (2008). Probability, Statistics and Random Processes for Electrical Engineering. Upper Saddle River, NJ: Pearson. ISBN 9780131471221.

[4] Pfeiffer, Paul E. (1978). Concepts of probability theory. Dover Publications. p. 18. ISBN 978-0-486-63677-1.

[5] Kallenberg, Olav (2002). Foundations of Modern Probability (2nd ed.). New York: Springer. p. 9. ISBN 0-387-94957-7.

[6] Foerster, Paul A. (2006). Algebra and Trigonometry: Functions and Applications, Teacher's Edition (Classics ed.). Upper Saddle River, NJ: Prentice Hall. p. 633. ISBN 0-13-165711-9.

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