Changes: Sample space

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In probability theory, the sample 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 $S$ , $\Omega$ , or $U$ (for "universal set").

For example, 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 {(M, M), (M, T), (T, M), (T, T)}. If the sample space is unordered, it becomes {{M, M}, {M, T}, {T, T}}.

For revealing a card from a Tech Box 5, 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).

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.

Examples

For instance, in the above trial of revealing the Tech card from a Daily Kit Box †, we have as a sample space $\Omega =\{M,A\}$ , with $s_{M}$ and $s_A$ being the events of receiving Mid-Tech and Advanced Tech, repsecctively.

Receiving Mid-Tech automatically means not receiving Advanced Tech and vice versa, so the outcomes are mutually exclusive.

Another possible sample space could be $\Omega _{2}=\{H\&R,H\&NR,T\&R,T\&NR\}$ . Here, $R$ stands for rains and $NR$ not rains.

Obviously, $\Omega_1$ is a better choice than $\Omega_2$ as we do not care about how the weather affects the tossing of a coin.

@@ Line 3: / Line 3: @@
 In [[probability theory]], the '''sample 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 <math>S</math>, <math>\Omega</math>, or <math>U</math> (for "universal set").
-For example, if the experiment is revealing the Tech card of 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 {(M, M), (M, T), (T, M), (T, T)}. If the sample space is unordered, it becomes {{M, M}, {M, T}, {T, T}}.
+For example, 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 {(M, M), (M, T), (T, M), (T, T)}. If the sample space is unordered, it becomes {{M, M}, {M, T}, {T, T}}.
-For revealing a card of a {{Tech Box 1}}, 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]]).
+For revealing a card from a {{Tech Box 5}}, 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]]).
 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 [[event]]s (a '''sigma-algebra''') and a [[probability]] assigned to each event (a '''probability measure''' function).
@@ Line 11: / Line 11: @@
 == Conditions of a sample space ==
 A set <math>\Omega</math> with outcomes <math>s_1, s_2, \ldots, s_n</math> ({{nobr|i. e.}} <math>\Omega = \{s_1, s_2, \ldots, s_n\}</math>) must meet some conditions in order to be a sample space:
-* The outcomes must be '''mutually exclusive''', {{nobr|i. e.}} if <math>s_j</math> takes place, then no other <math>s_i</math> will take place, <math>\forall i,j=1,2,\ldots ,n\quad i\neq j</math>.
+# The outcomes must be '''mutually exclusive''', {{nobr|i. e.}} if <math>s_j</math> takes place, then no other <math>s_i</math> will take place, <math>\forall i,j=1,2,\ldots ,n\quad i\neq j</math>.
-* The outcomes must be '''collectively exhaustive''', {{nobr|i. e.}}, on every experiment (or random trial) there will always take place some outcome <math>s_i \in \Omega</math> for <math>i \in \{1, 2, \ldots, n\}</math>.
+# The outcomes must be '''collectively exhaustive''', {{nobr|i. e.}}, on every experiment (or random trial) there will always take place some outcome <math>s_i \in \Omega</math> for <math>i \in \{1, 2, \ldots, n\}</math>.
-* The sample space (<math>\Omega</math>) 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.
+# The sample space (<math>\Omega</math>) 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.
-<!--
+== '''Examples''' ==
-For instance, in the trial of tossing a coin, we could have as a sample space <math>\Omega_1 = \{H,T\}</math>, where <math>H</math> stands for ''heads'' and <math>T</math> for ''tails''. Another possible sample space could be <math>\Omega_2 = \{H\&R, H\&NR, T\&R, T\&NR\}</math>. Here, <math>R</math> stands for ''rains'' and <math>NR</math> ''not rains''. Obviously, <math>\Omega_1</math> is a better choice than <math>\Omega_2</math> as we do not care about how the weather affects the tossing of a coin.
+For instance, in the above trial of revealing the Tech card from a {{Daily Kit Box}}, we have as a sample space <math>\Omega = \{M, A\}</math>, with <math>s_M</math> and <math>s_A</math> being the events of receiving Mid-Tech and Advanced Tech, repsecctively.
--->
+# Receiving Mid-Tech automatically means ''not'' receiving Advanced Tech and vice versa, so the outcomes are mutually exclusive.
+#
+Another possible sample space could be <math>\Omega_2 = \{H\&R, H\&NR, T\&R, T\&NR\}</math>. Here, <math>R</math> stands for ''rains'' and <math>NR</math> ''not rains''.
+Obviously, <math>\Omega_1</math> is a better choice than <math>\Omega_2</math> as we do not care about how the weather affects the tossing of a coin.
 == '''See also''' ==