Sample space
Set of all possible outcomes of an experiment or random trial.
In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. It is a fundamental component of a probabilistic model, along with an event space and a probability measure function.
- field
- Probability theory
- also_known_as
- Sample description space, possibility space, outcome space
- common_labels
- S, Ω, U (for universal set)
- elements_can_be
- Numbers, words, letters, or symbols
- types
- Finite, countably infinite, or uncountably infinite
Lore & Background
A sample space is denoted using set notation, with possible ordered outcomes listed as elements. For example, tossing a single coin yields the sample space {H, T}, where H means heads and T means tails. Tossing two coins gives {HH, HT, TH, TT}. A subset of the sample space is an event; if the outcome of an experiment is included in that subset, the event has occurred.
Reader's Guide
The sample space is one of three components in a probabilistic model, alongside a well-defined set of possible events (often the power set for discrete spaces or a σ-algebra for continuous spaces) and a probability assigned to each event. Conditions for a valid sample space include mutually exclusive outcomes (if one outcome occurs, no other does) and collectively exhaustive outcomes (every trial yields some outcome in the space). The sample space must have the right granularity, removing irrelevant information. For many experiments, multiple plausible sample spaces exist depending on the experimenter's interest; for instance, drawing a card could use ranks, suits, or both. When outcomes are equally likely, each of N outcomes is assigned probability 1/N.
Did You Know?
- A sample space can be represented visually by a rectangle, with outcomes denoted by points inside.
- Events may be represented by ovals enclosing points within the rectangle.
- For tossing a single six-sided die, the sample space is {1, 2, 3, 4, 5, 6}.
- The sample space for tossing two coins includes the ordered outcomes HH, HT, TH, and TT.
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