例 1
Listed below are the types and ages of four monkeys procured by a laboratory for a drug trial.

$\begin{array}{ccc} \hline \mbox{Monkey}&\mbox{Type}&\mbox{Age} \\ \hline 1&\mbo... ...box{Rhesus}&8 \\ 3&\mbox{Spider}&6 \\ 4&\mbox{Spider}&6 \\ \hline \end{array}$

Suppose two monkeys will be selected by lottery and assigned to an experimental drug. Considering all possible choices of two monkeys, make a Venn digram and show the following events.

$\begin{array}{ll} A:& \mbox{The selected monkeys are of the same type.} \\ B:& \mbox{The selected monkeys are of the same age.} \\ \end{array}$

Here the elementary outcomes are the possible choices of a pair of numbers from $\{1,2,3,4\}$. These pairs are listed and labeled as e1,e2,e3,e4,e5,e6for ease of reference.

$\{1,2\}\quad (e_1)$          $\{2,3\} \quad (e_4)$
$\{1,2\}\quad (e_1)$          $\{2,3\} \quad (e_4)$
$\{1,2\}\quad (e_1)$          $\{2,3\} \quad (e_4)$

The pair $\{1,2\}$ hasboth monkeys of the same type, and so does the pair $\{3,4\}$. Consequently, $A=\{e_1,e_6\}$. Those with the same ages are $\{1,3\}$, $\{1,4\}$, and $\{3,4\}$, so $B=\{e_2,e_3,e_6\}$. And the diagram.

例 2
Refer to the previous experiment of selectng two monkeys out of four. Let $A=[\mbox{same type}]$, $B=[\mbox{same age}]$, $C=[\mbox{different type}]$, Give the compositions of the events
$C, \overline{A}, A \cup B, AB,BC$

The pairs consisting of different types are $\{1,3\}$, $\{1,4\}$, $\{2,3\}$, and $\{2,4\}$, so $C=\{e_2,e_3,e_4,e_5\}$. The event $\overline{A}$ is the same as the event C. Employing the definitions of union and intersection, we obtain

$\begin{array}{rcl} A\cup B&=&\{e_1,e_2,e_3,e_6\} \\ AB&=&\{e_6\} \\ BC&=&\{e_2,e_3\} \\ \end{array}$