Home page for accesible maths 2.1 Events and the sample space 2.1.2 Events 2.2 Classical probability

Style control - access keys in brackets

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

2.1.3 Operations on events

Two related operations on events are intersection corresponding to ‘and’, and union corresponding to ‘or’.

The union ${A\cup B}$ of the events ${A}$ and ${B}$ is the set of outcomes $\omega$ that are in ${A}$ or in ${B}$ or in both. The intersection ${A\cap B}$ is the set of outcomes $\omega$ that are in ${A}$ and in ${B}$ .

The events ${A}$ and ${B}$ are mutually exclusive or disjoint if they have no outcomes in common; that is ${A\cap B}$ is the impossible event or equivalently ${A\cap B=\emptyset}$ .

Example 2.12.

Let ${A}$ be the event that a person has normal diastolic blood pressure (DBP) reading {DBP ${<90}$ }, and let ${B}$ be the event that person has borderline DBP readings { ${90\leq}$ DBP ${\leq 95}$ }.

The event ${A\cup B=\{{\color[rgb]{1,1,1}\quad\mbox{DBP}\leq 95}\quad\}}$ and ${A\cap B=\emptyset}$ is the empty set, or impossible event.

Exercise 2.13.

A coin is thrown twice: let ${A}$ denote the event that the same face occurs twice, and let ${B}$ denote the event that the coin shows different faces.

Solution.

That is ${A=\{{\color[rgb]{1,1,1}\quad\mbox{HH,TT}}\quad\}}$ and ${B=\{{\color[rgb]{1,1,1}\quad\mbox{HT,TH}}\quad\}}$ and so ${A\cap B=\emptyset}$ .

What does it mean to be mutually exclusive?
Let $A\cap B=\emptyset$ — there are no sample points in both $A$ and $B$ .
Suppose that $A$ occurs, meaning $\omega\in A$ .
Therefore $\omega\notin B$ — $B$ does not occur.
Similarly, if $B$ occurs then $A$ does not occur.
Mutually exclusive means at most one of $A$ and $B$ can occur.

The complementary event to ${A}$ is the event ${A^{c}}$ consisting of those outcomes that are in ${\Omega}$ but are not in ${A}$ . Note that ${A\cup A^{c}=\Omega}$ and ${A\cap A^{c}=\emptyset}$ .

Example 2.14.

If ${A}=\{2,4,6\}$ is the event that the die shows an even face then ${A^{c}={\color[rgb]{1,1,1}\quad\{1,3,5\}}}$ and is identical to the event that the die shows an odd face.

A partition of the sample space splits the sample space into disjoint subsets.

Example 2.15.

The score on a die can be partitioned according to whether it is even or odd: ${\{1,2,3,4,5,6\}=}$ ${\{2,4,6\}\cup\{1,3,5\}}$ .
Write ${A_{1}=\{2,4,6\}}$ and ${A_{2}=\{1,3,5\}}$ . Then ${A_{1}}$ and ${A_{2}}$ are mutually exclusive and exhaustive.
We may express the sample space as ${\Omega=A_{1}\cup A_{2}}$ where ${A_{1}\cap A_{2}=\emptyset}$ .

More generally, the ${k}$ sets ${A_{1},A_{2},\dots,A_{k}}$ form a partition of the set ${B}$ if the sets ${A_{1},A_{2},\dots,A_{k}}$ are mutually exclusive and exhaustive, so that $\displaystyle A_{i}\cap A_{j}=\emptyset,$ for all $i\neq j$ and $\displaystyle B=A_{1}\cup A_{2}\cup\dots\cup A_{k}.$