Exhaustive and Complementary Events

Exhaustive Events

When a sample space $S$ is partitioned into some mutually exclusive events such that their union is the sample space itself, then the events are called exhaustive events or collective events.

Suppose a die is tossed and the sample space is

Let       $A = \left\{ {1,2} \right\}$        $B = \left\{ {3,4,5} \right\}$         $C = \left\{ 6 \right\}$

Hence the events $A,B$ and $C$ are mutually exclusive because $A \cap B \cap C = \phi$ and $A \cup B \cup C = S$. As shown in the figure, the three events $A,B$ and $C$ are exhaustive.

$A \cap B \cap C = \phi$ and $A \cup B \cup C = S$

Complementary Events

If $A$ is an event defined in the sample space $S$, then $S - A$ is denoted by $\overline A$ and is called a complement of $A$.

Thus,
$\overline A = S - A$   or   $A \cup \overline A = S$

The figure shows the event $A$ and the complement of $A$.