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In probability theory, the **sample space** or **event 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 (with curly brackets), and the possible ordered outcomes are listed as elements in the set. It is common to refer to a sample space by the labels $ \Omega $, $ S $, or $ U $ (for "universal set").

Examples:

- If the experiment is revealing the Tech card from a
**Daily Kit Box †**which grants either Mid-Tech or Advanced Tech, the sample space is typically the set {Mid-Tech, Advanced Tech} or {M, T}. For revealing the Tech card of two Daily Kit Boxes, the corresponding sample space would be the set of (ordered) tuples {(M, M), (M, T), (T, M), (T, T)}. If the sample space is unordered, it becomes the set of (unordered) sets {{M, M}, {M, T}, {T, T}}. - For revealing a card from a
**Tech Box 2**, the typical sample space is {E, I, M, A} (Early, Initial, Mid- and Advanced Tech, as Tech Boxes do not grant bike Tech since the 2019 Spring Update). - The description of the Volkswagen I.D. R
**Festival Coin Pack**says "Contains up to 8 Festival Coins!", but in reality, the only possible outcomes are 3, 5 or 8 Coins. Thus, the sample space is {**3**,**5**,**8**} or short {3, 5, 8}.

A well-defined sample space is one of three basic elements in a probabilistic model (a **probability space**); the other two are a well-defined set of possible events (a **sigma-algebra**) and a probability assigned to each event (a **probability measure** function).

## Conditions of a sample space

A set $ \Omega $ with outcomes $ s_1, s_2, \ldots, s_n $ (i. e. $ \Omega = \{s_1, s_2, \ldots, s_n\} $) must meet some conditions in order to be a sample space:

- The outcomes must be
**mutually exclusive**, i. e. if $ s_j $ takes place, then no other $ s_i $ will take place, $ \forall i,j=1,2,\ldots ,n\quad i\neq j $. - The outcomes must be
**collectively exhaustive**, i. e., on every experiment (or random trial) there will always take place some outcome $ s_i \in \Omega $ for $ i \in \{1, 2, \ldots, n\} $. - The sample space ($ \Omega $) must have the
**right granularity**depending on what we are interested in. We must remove irrelevant information from the sample space. In other words, we must choose the right abstraction.

## **Example**

For instance, in the above trial of revealing a card from a ** Tech Box 2**, we have as a sample space $ \Omega = \{E, I, M, A\} $.

- Receiving Early Tech automatically means
*not*receiving any other Tech; this goes for all Tech cards, so the outcomes are mutually exclusive. - Every outcome of revealing a card will be an element of $ \Omega = \{E, I, M, A\} $, so they are collectively exhaustive.
- The smallest oberved unit is a card. We could choose the whole box as observed unit, but this would enlarge $ \Omega $ to $ \{\{E, E\}, \{E, I\}, \{E, M\}, \ldots, \{M, A\}, \{A, A\}\} $. This would make calculations more complex, and it would be difficult to compare the outcomes with those of opening a
**Tech Box 5**. A comparison with the**Daily Kit Box †**would even be impossible because its cards have three different sample spaces (see example).