# Mutually Exclusive Events

If two events have no elements in common (Their intersection is the empty set.), the events are called mutually exclusive.   Thus, $P\left(A\cap B\right)=0$ .  This means that the probability of event $A$ and event $B$ happening is zero.  They cannot both happen.

Example 1:

A pair of dice is rolled.  The events of rolling a $5$ and rolling a double have NO outcomes in common so the two events are mutually exclusive.
A pair of dice is rolled.  The events of rolling a $4$ and rolling a double have the outcome $\left(2,2\right)$ in common so the two events are not mutually exclusive.

Example 2:

From a group of $6$ freshmen and $5$ sophomores, $3$ students are to be selected at random to form a committee.  What is the probability that at least $2$ freshmen are selected?

The committee will have at least $2$ freshmen if either $2$ freshmen and $1$ sophomore are selected (event $A$ )or $3$ freshmen are selected (event $B$ ).   Since the two events are mutually exclusive

$\begin{array}{l}P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)\hfill \\ \begin{array}{l}P\left(A\right)=\frac{{}_{6}{}^{}C{}_{2}\cdot {}_{5}{}^{}C{}_{1}}{{}_{11}{}^{}C{}_{3}}=\frac{15\cdot 5}{165}=\frac{75}{165}=\frac{5}{11}\\ \end{array}\hfill \\ \begin{array}{l}P\left(B\right)=\frac{{}_{6}{}^{}C{}_{3}}{{}_{11}{}^{}C{}_{3}}=\frac{20}{165}=\frac{4}{33}\\ \end{array}\hfill \\ P\left(A\cup B\right)=\frac{5}{11}+\frac{4}{33}=\frac{15}{33}+\frac{4}{33}=\frac{19}{33}\hfill \end{array}$