kristin is collecting yu-gi-oh! cards. each day she gets one uniformly random card out of n possible card types. what is the expected number of days until she has collected at least one copy of each of the n card types?

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31-10-2022
Mathematics

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kristin is collecting yu-gi-oh! cards. each day she gets one uniformly random card out of n possible card types. what is the expected number of days until she has collected at least one copy of each of the n card types?

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contexto1025 contexto1025 · Answer 1 · 2022-11-09T06:32:58-05:00

Using the concepts of complex numbers, we got that Kristin requires n days at least to get one copy of each of the n card types.

Complex numbers are those numbers that are expressed in the form of a + ib where, a, b are real numbers and ‘i’ is an imaginary number known as “iota”. The value of i = (√-1). Like, 2+3i is an complex number, where 2 is a real number (Re) and 3i is a imaginary number (Im).

So, let suppose Z be the number of days until she collects at least one copy of each the n card types. For each i let [tex]Y_i[/tex] be the days she gets her ith type of card. In particular, Y₁=1 and

Y₁<Y₂<.........Yₙ=Z.

Let [tex]Z_{i}[/tex]=[tex]Y_i-Y_(i-1)[/tex].

Then Z=Z₁+Z₂+Z₃+.........Zₙ.

Identifying each [tex]Z_i[/tex] as a geometric progression and use this to find E(|Z|).

This will tell that she requires at least n days.

Hence, the expected number of days so that Kristin collected at least one copy of the n card types is n days.

To know more about complex numbers, visit here:

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