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
\[S = \left\{ {1,2,3,4,5,6} \right\}\]

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 $$A$$.

$$\overline A = S – A$$   or   $$A \cup \overline A = S$$

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