Discrete Mathematics
# Expected Value

There are seven balls of the same shape in a pocket, each labelled with an integer from 0 to 6. Danielle draws a ball out, takes the number written on it as $x,$ and puts the ball back in. Then she makes another draw, and takes the number on it as $y.$ If the numbers $x$ and $y$ satisfy $\sin\left(\frac{(x+y)}{6}\pi\right)=\frac{\sqrt{3}}{2}\ x,y\in[0,6]$ what is the conditional expectation of $x?$

**Note:** Every ball has a different number written on it, i.e. integers from 0 to 6 are all used once.