Mutually Exclusive Events

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

$\begin{array}{l}P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)\hfill \\ \begin{array}{l}P\left(A\right)=\frac{{}_6{}^{}C{}_2\cdot {}_5{}^{}C{}_1}{{}_{11}{}^{}C{}_3}=\frac{15\cdot 5}{165}=\frac{75}{165}=\frac{5}{11}\\ \end{array}\hfill \\ \begin{array}{l}P\left(B\right)=\frac{{}_6{}^{}C{}_3}{{}_{11}{}^{}C{}_3}=\frac{20}{165}=\frac{4}{33}\\ \end{array}\hfill \\ P\left(A\cup B\right)=\frac{5}{11}+\frac{4}{33}=\frac{15}{33}+\frac{4}{33}=\frac{19}{33}\hfill \end{array}$