Not Mutually Exclusive Events

Two events are called not mutually exclusive if they have al least one outcome common between them. If the two events A and B are not mutually exclusive events, then A \cap B \ne \phi .  Similarly, A,B and C are not mutually exclusive events if A \cap B \cap C \ne \phi . Thus they must have at least one common point between them.

Consider a sample space:

S  = \left\{ {1,2,3,4,5,6,7,8,9,10,11} \right\}

Let A = \left\{ {2,3,5,7,11} \right\}   and   B = \left\{ {1,3,5,7,9,11} \right\}

Here A \cap B = \left\{ {3,5,7,11} \right\}

Thus, A \cap B \ne \phi i.e. A \cap B exist. Here A and B are not mutually exclusive events. A \cap B consists of outcomes which are common to both A and B. As shown in the figure a Vann diagram in which A and B are not mutually exclusive events. Some area under A is common with B. If the event A is a part of the event B, then A  \cap B = A. This is shown in the figure:


not-mutually-exclusive-events

A \cap B


not-mutually-exclusive-events2

A \cap B = A

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