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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.

# Level 4

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 $$\text{unit}$$)?

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?

Sarah the squirrel is trying to find her acorn, but she can't remember where she left it! She starts in the lower-left corner of the $$2\times 2$$ grid, and at each point, she randomly steps to one of the adjacent vertices (so she may accidentally travel along the same edge multiple times). What is the expected value for the number of steps Sarah will take before she finds her acorn in the top-right corner?

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 $$\lfloor 1000E\rfloor$$?

A fair coin is tossed repeatedly, until 5 consecutive heads occur. What is the expected number of coin tosses?

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