Discrete Mathematics
# Expected Value

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 \( p = \left( \cos ( 2 \pi \theta) , \sin ( 2 \pi \theta) \right) \).

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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