Events as Subsets of the Sample Space

Events as Subsets of the Sample Space

Once the sample space $S$ of a random experiment is fixed, an event is defined formally as any subset of $S$. This is exactly the set-theoretic language from the Sets chapter, now applied to probability.

For a die thrown once, $S = \{1,2,3,4,5,6\}$. The event "an even number turns up" is $E = \{2, 4, 6\}$, a $3$-element subset of $S$. If the die actually shows a $4$, the event $E$ has occurred, because $4 \in E$.

Two Special Events

• The sure event (or certain event) is $S$ itself — since every outcome belongs to $S$, this event occurs on every trial.
• The impossible event is the empty set $\phi$ — since no outcome belongs to $\phi$, it can never occur.

Simple and Compound Events

An event containing exactly one sample point, such as $\{3\}$, is called a simple (or elementary) event. An event containing two or more sample points, such as $\{2, 4, 6\}$, is called a compound event.

Algebra of Events — 'Not', 'And', 'Or'

Because events are sets, the usual set operations translate directly into everyday event language: