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There are 100 types of cards, each type with a number from 1 to 100 on it. Each minute I am given a random card. Random means that the number of my card has an equal chance of being 1,2,3,...,100. I stop when I have cards of each number, that is, I stop collecting once I have each number on at least one card. How many minutes is this process expected to take? That is, find the expected value of the number of minutes I have to wait.

E.g. I could potentially get duplicates so this method will take a lot more than 100 minutes. Once I get 99 different cards, I will stop as soon as I get the other one which I haven't got yet.

Thanks for helping everyone!


marked as duplicate by Did probability Oct 1 '15 at 7:33

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  • $\begingroup$ Why not start small, with 1, 2, 3,... cards, so as to get some insight into what is happening ? $\endgroup$ – true blue anil Oct 1 '15 at 7:00
  • $\begingroup$ Does the source of the cards have infinite capacity, i.e. say if the first $100$ cards are marked $1$, then for the rest of the time can we see any more card marked $1$? $\endgroup$ – Samrat Mukhopadhyay Oct 1 '15 at 7:10
  • $\begingroup$ This is called the coupon collector's problem and has already many duplicates on the site. $\endgroup$ – Did Oct 1 '15 at 7:35

Well the first card 1 minutes, the second card $\frac{100}{99}$ minutes, the third card $\frac{100}{98}$ minutes, and so on. So it is expected to take $(1 + \frac{100}{99} + \frac{100}{98} + ... + \frac{100}{1})$ which is around 519 minutes.


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