Sample Space and Events

Compact study note.

Summary

One sample space lists possible outcomes; events are sets of outcomes whose probability is defined. This distinction prevents many notation errors in later probability rules.^[1]

Prerequisites

• Prerequisites: None

Notation and Assumptions

$\mathrm{\Omega }$ is the sample space. An outcome is $\omega \in \mathrm{\Omega }$ . An event is one subset $A\subseteq \mathrm{\Omega }$ that belongs to the event collection $\mathcal{F}$ .

Essential Result

For finite elementary models, every subset of $\mathrm{\Omega }$ can be an event. For infinite models, events must be chosen through one sigma-algebra.

Small Example

For two coin flips, $\mathrm{\Omega }=\{HH,HT,TH,TT\}$ . The event 'exactly one head' is $A=\{HT,TH\}$ , not one single outcome.

Common Mistakes

• Writing $P(H)$ before defining $H$ as an event set.
• Assuming every subset of an uncountable sample space is automatically measurable.

Connections

• Probability
• Sigma-Algebra
• Probability Space

References