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.