# Not Mutually Exclusive Events

Two events are called not mutually exclusive if they have at least one outcome in common. 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 $$B$$. The figure below shows a Venn diagram in which $$A$$ and $$B$$ are not mutually exclusive events. An area under $$A$$ is common with $$B$$. If the event $$A$$ is a part of event $$B$$, then $$A \cap B = A$$. This is shown in the figure:

\[A \cap B\]

\[A \cap B = A\]