In
probability theory
Probability theory 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 ...
, an outcome is a possible result of an
experiment
An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into cause-and-effect by demonstrating what outcome occurs whe ...
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 (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
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 a ...
. 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
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
). 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
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
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
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
). 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
Dice (singular die or dice) are small, throwable objects with marked sides that can rest in multiple positions. They are used for generating random values, commonly as part of tabletop games, including dice games, board games, role-playing ...
, 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
*
*
*
*
*
References
External links
*{{Commonscat-inline
Experiment (probability theory)