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 collectively 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 three events A,B and C which are exhaustive.


exhaustive-events

 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 complement of A.

Thus,
                        \overline A = S - A   or   A \cup  \overline A = S
In the figure shown that the event A and the complement of A.


complement-event

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