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The problem is like this:

A standard deck contains $52$ cards, $4$ each of $2,3,4,5,6,7,8,9,J,Q,K,A$. Now start the following process. Pick a random card from the deck, show it, and then return it to the deck. Continue repeating this process, stopping when each type of card, $2,3,4,5,6,7,8,9,J,Q,K,A$ has been seen at least once. What is the expected number of cards that you will have drawn?

My idea is that as each type of card have the same number of cards ($4$), the problem can be converted into a deck wich contains $13$ cards only with $1-13$. However, I have no idea about the next step.

joriki
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David Wong
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1 Answers1

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This is the Coupon collectors problem. The answer for $N=13$ is $42$ as can be seen from the attached table. The true expectation is given by $$E_{13} = 13 H_{13} = 13\cdot(1+1/2+1/3+1/4+1/5+1/6+1/7+1/8+1/9+1/10+1/11+1/12+1/13) \\ \approx 41.34$$

AlexR
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