= Sample space
{title2=$\Omega$}
The <sample space> is the <set> $\Omega$ of outcomes represented by a <probability> model. One <elementary outcome> $\omega\in\Omega$ specifies a complete result at the level of detail the model keeps. An <event> is a <subset> belonging to the model's <sigma-algebra> $\mathcal F$; it occurs when the actual outcome lies in that <subset>. Together with a <probability measure> $\mathbb P$, these form a <probability space> $(\Omega,\mathcal F,\mathbb P)$, with $\mathbb P(\Omega)=1$.<Introduction to Probability, Statistics, and Random Processes by Hossein Pishro-Nik>[§1.3.1 Random Experiments: definitions of outcome, sample space, and event; three-toss example]{ref}
Choosing a <sample space> means deciding which distinctions matter. For two tosses, recording the ordered head-or-tail sequence distinguishes $HT$ from $TH$; recording only the number of heads merges them. Neither representation is automatically wrong, but the second cannot answer a question about which toss was heads. Keeping a detailed <sample space> and defining a <random variable> for the desired observation lets several questions share one model.
A <sample space> alone assigns no <probabilities>. Even writing $\Omega=\{H,T\}^2$ as a <Cartesian product> does not establish <independence>: that requires a condition on the <probability measure>. On a finite <sample space> with all <subsets> admitted as <events>,
$$
\mathbb P(A)=\sum_{\omega\in A}\mathbb P(\{\omega\}).
$$
Only when all elementary <probabilities> are equal does this become $\mathbb P(A)=|A|/|\Omega|$, where the bars denote <cardinality>. This is the <uniform distribution on a finite set>, an additional modelling choice.
<Sample spaces> can also be infinite. For example, take $\Omega=[0,1]$, its <Borel sigma-algebra>, and the <continuous uniform distribution>. Then $\mathbb P([a,b])=b-a$ for $0\leq a\leq b\leq1$, so each <singleton> has <probability> zero while $\mathbb P(\Omega)=1$. Thus an <elementary outcome> need not carry positive <probability>, and a continuous model cannot be understood by counting equally likely points. The distinction is developed in <zero-probability event> and <countable additivity>.
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