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Discrete Mathematics

Expected Value

Expected Value: Level 4 Challenges

         

Each sealed Storelings pack contains a single Storeling toy chosen at random. There are 3 distinct Storelings toys in the set, and each toy has an equal chance to be in each pack. Also, each pack is independent of every other pack.

What is the expected value of the number of packs one would need to open in order to obtain at least one copy of each toy?

Consider the unit circle \( x^2 + y^ 2 = 1 \).
Choose 3 points uniformly at random on the circumference, which divides the circle into 3 arcs.

What is the expected length of the arc that contains the point \( (1,0) \)?


Technical details: We pick a point on the circumference uniformly at random in the following manner.
1. First select \( \theta \sim U[0,1] \), the uniform distribution on the unit interval
2. Next, we pick the point \( p = \left( \cos (\theta \pi) , \sin (\theta \pi) \right) \).

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?

12 zombies are in a room. 6 are green. 6 are blue.

Every minute they randomly form four groups of three. In any group, all three become the color of the majority (i.e. if two blue zombies and one green zombie are in a group, then they all become blue).

The expected number of minutes before all zombies are the same color can be written as \(\dfrac{a}{b}\), where \(a\) and \(b\) are coprime positive integers. What is \(a+b\)?


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

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