How many random samples needed to pick all elements of set?

Question

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.

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 09:33
(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 09:41
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 09:50
I guess for infinite sets the answer is trivial unless you don't distinguish infinite cardinalities. — user123454321, Jan 24 '13 at 09:51
Asaf Karagila, the definition of random is defined by the behavior of my system's PRNG, which I assume to be close to uniform. — user59475, Jan 24 '13 at 10:04

score 12 · Accepted Answer · answered Jan 24 '13 at 10:01

12

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

answered Jan 24 '13 at 10:01

MJD

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Thanks, this is exactly what I was looking for. – user59475 Jan 24 '13 at 10:07
For more details about this, see https://math.stackexchange.com/a/4503130/25554 – MJD May 22 '23 at 11:09

How many random samples needed to pick all elements of set?

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