Home page for accesible maths 2.1 Events and the sample space 2.1.1 The sample space,

{\Omega}

Style control - access keys in brackets

Font (2 3) - + Letter spacing (4 5) - + Word spacing (6 7) - + Line spacing (8 9) - +

2.1.2 Events

Sets of outcomes, rather than outcomes themselves, are often of interest.

An event ${A}$ is a subset of the possible outcomes contained in the sample space ${\Omega}$ . We write ${A\subseteq\Omega}$ to denote that every element of ${A}$ is a member of ${\Omega}$ . An event ${A\subseteq\Omega}$ occurs if, when the experiment is performed, the outcome ${\omega\in\Omega}$ satisfies ${\omega\in A}$ .

Example 2.9.

A die is rolled, so the sample space is $\Omega=\{1,2,3,4,5,6\}$ . The event that the die shows an even face is the subset ${\{\mbox{2,4,6}\}}$ .

Example 2.10.

Two coins are tossed, so the sample space is $\Omega=\{HH,HT,TH,TT\}$ . The event that a tail occurs is the subset ${\{\mbox{HT, TH, TT}\}}$ .

Example 2.11.

Illustrate the sample space when two dice are thrown and indicate the event that the sum is 5.

${\begin{array}[]{r|c|c|c| c|c|c|}6&.&.&.&.&.&.\\ \hline 5&.&.&.&.&.&.\\ \hline 4&*&.&.&.&.&.\\ \hline 3&.&*&.&.&.&.\\ \hline 2&.&.&*&.&.&.\\ \hline 1&.&.&.&*&.&.\\ \hline&1&2&3&4&5&6\\ \end{array}}$

Special events

There are some special events which exist for any sample space:

•

the whole sample space $\Omega$ is an event; when the experiment occurs, we observe an $\omega\in\Omega$ , so the event $\Omega$ is certain to occur.
•

the empty set ${\emptyset=\{\}}$ is an event with no elementary outcomes; $\omega\notin\emptyset$ for any $\omega$ , so $\emptyset$ is an event which will never occur.