Outcome (probability)
A possible result of an experiment or trial.
In probability theory, an outcome is one possible result from an experiment or trial. Every outcome in a given experiment is a distinct random element, and outcomes are mutually exclusive—only one can happen per trial. The complete set of all possible outcomes is called the sample space. For instance, flipping a coin twice yields four outcomes: (H, H), (H, T), (T, H), and (T, T), where H stands for heads and T for tails.
Outcomes should not be confused with events. An event is a set—or informally, a group—of outcomes. For example, the event “at least one heads” includes every outcome in the sample space except (T, T). Because individual outcomes are often not useful on their own, or because there may be too many (even infinitely many) to handle, outcomes are grouped into events that satisfy certain conditions. The collection of all such events forms a sigma-algebra. An event containing exactly one outcome is an elementary event, while the event containing all outcomes is the sample space itself. A single outcome can belong to many different events.
When the sample space is finite, any subset of it can be an event (meaning the entire power set is used). But this fails for uncountably infinite sample spaces—for example, when outcomes are real numbers. In such cases, defining a probability space often requires excluding some subsets from being events.
Outcomes have probabilities between zero and one. In a finite discrete distribution, each outcome gets its own probability. In a continuous distribution, individual outcomes have zero probability; only ranges of outcomes can have non-zero probabilities. Some “mixed” distributions combine continuous stretches with discrete outcomes (called atoms), which can have non-zero probabilities. Under the measure-theoretic definition of a probability space, the probability of an outcome may not even be defined, because the set of events (a sigma-algebra) may not be the full power set.
In some sample spaces, it is reasonable to assume all outcomes are equally likely. For example, a fair coin toss gives heads and tails with equal probability. This assumption underlies most randomization in games of chance—rolling dice, shuffling cards, spinning tops, drawing lots—though players may cheat by introducing bias (e.g., marked cards, loaded dice). Some probability treatments always define outcomes to be equally likely, but not all experiments fit this pattern. Tossing a thumbtack, for instance, has two outcomes (point up or down) with no symmetry to suggest equal likelihood.
- field
- Probability theory
- known_for
- Defining the basic element of a sample space in probability experiments
- related_concepts
- Event, sample space, probability distribution, probability space
Lore & Background
In probability theory, an outcome is a possible result of a single experiment or trial. Each outcome is a unique random element, and different outcomes are mutually exclusive, meaning only one outcome can occur on each trial. The complete set of all possible outcomes for an experiment forms the sample space. For instance, when flipping a coin twice, the sample space consists of four outcomes: (H, T), (T, H), (T, T), and (H, H). Outcomes are distinct from events, which are sets or groups of outcomes. An event containing exactly one outcome is called an elementary event. The event that contains every possible outcome is the sample space itself. A single outcome may belong to many different events. In finite sample spaces, any subset of the sample space can be an event, but this is not always possible when the sample space is uncountably infinite, such as when outcomes are real numbers. In such cases, certain subsets must be excluded from being events when defining a probability space. Outcomes have probabilities between zero and one. In discrete distributions with a finite sample space, each outcome is assigned a specific probability. In continuous distributions, individual outcomes have zero probability, and non-zero probabilities apply only to ranges of outcomes. Some mixed distributions contain discrete outcomes, called atoms, which can have non-zero probabilities. Under the measure-theoretic definition of a probability space, the probability of an individual outcome need not even be defined. In some sample spaces, outcomes are assumed equally likely, as with a fair coin toss, but this assumption does not hold for all experiments, such as tossing a thumb tack.
Reader's Guide
Outcomes are central to probability theory as they form the sample space of an experiment. They are mutually exclusive and each trial yields exactly one outcome. While individual outcomes may be of little practical interest, they are grouped into events, which are sets of outcomes satisfying some condition. In finite sample spaces, any subset can be an event, but in uncountably infinite sample spaces, some subsets may be excluded. Outcomes may have probabilities between zero and one; in discrete distributions each outcome has a probability, while in continuous distributions individual outcomes have zero probability. Some distributions mix continuous and discrete outcomes, where discrete outcomes are called atoms. Under the measure-theoretic definition, the probability of an outcome need not be defined. Equally likely outcomes are assumed in many randomization tools, but not all experiments are easily described by equally likely outcomes.
Did You Know?
- Each possible outcome of a particular experiment is a unique random element.
- Different outcomes are mutually exclusive; only one outcome will occur on each trial.
- In a continuous distribution, individual outcomes all have zero probability.
- Under the measure-theoretic definition of a probability space, the probability of an outcome need not be defined.
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