In probability theory, an outcome is a possible result of an experiment or trial. 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.
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.
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.
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.
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 
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. 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.
See also
- Event (probability theory) – In statistics and probability theory, set of outcomes to which a probability is assigned
- Sample space – Set of all possible outcomes or results of a statistical trial or experiment
- Probability distribution – Mathematical function for the probability a given outcome occurs in an experiment
- Probability space – Mathematical concept
- Realization (probability) – Observed value of a random variable
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.
External links
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Media related to Outcome (probability) at Wikimedia Commons
In probability theory an outcome is a possible result of an experiment or trial 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 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 eventsSince 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 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 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 outcomeOutcomes 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 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 s algebra on S displaystyle S and not necessarily the full power set Equally likely outcomesFlipping 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 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 See alsoEvent probability theory In statistics and probability theory set of outcomes to which a probability is assigned Sample space Set of all possible outcomes or results of a statistical trial or experiment Probability distribution Mathematical function for the probability a given outcome occurs in an experiment Probability space Mathematical concept Realization probability Observed value of a random variableReferences 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 External linksMedia related to Outcome probability at Wikimedia Commons
