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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.

# Expected Value: Level 4 Challenges

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?

For fixed integers $$n$$ and $$k$$, find all $$k$$-tuples of non-negative integers $$(a_1, a_2, \ldots a_k)$$ such that

$a_1 + a_2 + \cdots + a_k = n$

For each integer $$i$$ from 0 to $$n$$, let $$p_i$$ be the probability that one of the $$a_j$$ is equal to $$i$$. What is the value of

$1\times p_1 + 2 \times p_2 + \cdots n \times p_n ?$

Two unit vectors in two-dimensional space $$\hat{\textbf{a}}$$ and $$\hat{\textbf{b}}$$ are added together. The expected magnitude of the resulting vector $$\textbf{a+b}$$ is equal to $$E.$$ What is $$\lfloor100E\rfloor?$$

Also try Daniel Liu's Expected Distance on a Circle.

###### Image credit: Wikipedia P.Fraundorf

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 $$\text{unit}$$)?