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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.

A square \(PQRS\) is inscribed inside a square \(ABCD\), with \(P \) on \(AB\). If \(P \) is uniformly distributed along side \(AB \), the expected area of \(PQRS\) is \(24\text{ unit}^2\). What is the side length of square \(ABCD\) (in \(\text{unit}\))?

Give your answer to 1 decimal place.

A game costs $150 to play. In this game, you roll a fair six-sided die until all six numbers have been rolled at least once. You are then paid 10 times the number of rolls you made.

For example, if the rolls were 3, 5, 4, 3, 2, 5, 1, 4, 1, 3, 6, then you would get \((10)(11) = 110\) dollars.

Including the price to play, what is your expected value in this game?

An angle \(\theta\) is chosen randomly in the interval \([0,2\pi )\), and its corresponding point on the unit circle is plotted. Let this point be \(P\). Now let \(A=(1,0)\) and \(B=(-1,0)\).

Let the expected value of \(PA+PB-2\) be \(E\).

What is \(\lfloor 1000E\rfloor\)?

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