# Mutually Exclusive Events

Two events are called mutually exclusive or disjointed if they do not have any outcome common between them. If the two events $$A$$ and $$B$$ are mutually exclusive, then $$A \cap B = \phi $$ (null set). For three mutually exclusive events $$A,B$$ and $$C$$, we have $$A \cap B \cap C = \phi $$.

Suppose there is a sample space $$S$$ as:

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

Let $$A = \left\{ {3,6,9} \right\}$$ and $$B = \left\{ {5,10} \right\}$$

Here $$A \cap B = \phi $$. Thus $$A$$ and $$B$$ are mutually exclusive events. Both $$A$$ and $$B$$ belong to the same sample space but they are completely different and both cannot happen at the same time. A class of students may contain first grade, second grade and third grade. When a student is selected from the class, he will be part of any one of the three groups of students. Thus three groups of students are disjointed or mutually exclusive.

When the two events $$A$$ and $$B$$ are mutually exclusive, we can show them with the help of a Venn diagram. The Venn diagram in the figure shows that $$A \cap B = \phi $$.

\[A \cap B = \phi \]