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This question already has an answer here:

If repeatedly picking a random element from a set, what is the expected number of times I'd have to pick before seeing all the elements of the set?

Edit: when picking an element, it is simply counted and not removed from the set, so it can be picked again.

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marked as duplicate by joriki, Shailesh, Daniel W. Farlow, Claude Leibovici, M. Vinay Jun 18 at 4:24

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If you can see only by picking the elements, shouldn't the expected number be the carnality of the set? – user123454321 Jan 24 '13 at 9:33
(Assuming you meant cardinality). No, I forgot to mention that the elements are not removed from the set when picked, they are only counted. – user59475 Jan 24 '13 at 9:41
Yeah that was a typo. Is the set finite? – user123454321 Jan 24 '13 at 9:50
I assume that you are talking about finite sets, right? Is there a particular distribution to how "random" works? – Asaf Karagila Jan 24 '13 at 9:50
I guess for infinite sets the answer is trivial unless you don't distinguish infinite cardinalities. – user123454321 Jan 24 '13 at 9:51
up vote 7 down vote accepted

This is the coupon collector's problem. The expected number of picks required to choose all the elements of the set is $$nH_n = n\sum_{i=1}^n\frac1i.$$

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Thanks, this is exactly what I was looking for. – user59475 Jan 24 '13 at 10:07

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