I have 5 marbles numbered 1 through 5 in a bag.  Suppose I take out two different marbles at random.  What is the expected value of the sum of the numbers on the marbles?
There are $\binom{5}{2} = 10$ different pairs of marbles can be drawn, and the expected value of the sum is the average of the sums of each pair.  This is  \begin{align*}
\frac{1}{10}((1+2)+(1+3)+(1+4)+(1+5)+(2+3)&\\
+(2+4)+(2+5)+(3+4)+(3+5)+(4+5))&=\frac{60}{10} = \boxed{6}. \end{align*}