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# Expected Value

If you play roulette, are you going to break-even? Maybe sometimes, but not if you play forever, because the expected value is negative. The expected value finds the average outcome of random events.

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.

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