## FANDOM

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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 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 , 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 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 {, , } 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:

1. The outcomes must be mutually exclusive, i. e. if $s_j$ takes place, then no other $s_i$ will take place, $\forall i,j=1,2,\ldots ,n\quad i\neq j$.
2. 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 $i \in \{1, 2, \ldots, n\}$.
3. 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 , we have as a sample space $\Omega = \{E, I, M, A\}$.

1. Receiving Early Tech automatically means not receiving any other Tech; this goes for all Tech cards, so the outcomes are mutually exclusive.
2. Every outcome of revealing a card will be an element of $\Omega = \{E, I, M, A\}$, so they are collectively exhaustive.
3. 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 . A comparison with the would even be impossible because its cards have three different sample spaces (see example).