Sample space

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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:

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 Tech Box 2, 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 Tech Box 5. A comparison with the Daily Kit Box † would even be impossible because its cards have three different sample spaces (see example).

See also

• Event (probability theory)
• Experiment
• Sample