# 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:

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: