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There are bags with n balls, and each ball can have any of the n different colors with equal probability. A bag can contain each color multiple times. What is the expected number of bags until we get at least one ball of each color?

The closest problem I could find is the coupon collector's problem, but in that problem you buy the boxes one by one, whereas in this problem you can only select a multiple of n balls.

What I got so far is the expected number if we are only interested in getting one specific color $i$, which would then have a geometric distribution with $p=1-(\dfrac{n-1}{n})^n$, since getting color $i$ in a bag would be the complement of getting n balls with a color other than $i$, but I feel like this not the right direction.

joriki
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Mustafa Enes Batur
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    Please edit to include your efforts. A natural starting point might be to work this out for small $n$. Also note that Linearity of Expectation is your friend here. I should also note that your title question and the question in the body are not the same. – lulu Mar 17 '23 at 13:54
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    Similar thinking to coupon collector gets you there. What is the chance the first ball is a new color? The second? The third? – Ross Millikan Mar 17 '23 at 13:56
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    How many bags are there? You seem to have left this out. Or are you perhaps asking for the expected number of bags until getting a bag with a ball of each color? Note, since there are $n$ colors and $n$ balls, the statement "at least one ball of each color" is the same as "exactly one ball of each color" – JMoravitz Mar 17 '23 at 14:04
  • I just edited the question, sorry if it was not clear. It is just a variation of the coupon collector's where you draw n coupons instead of just 1 at each step. – Mustafa Enes Batur Mar 17 '23 at 14:22
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    This is a special case of the question I closed this as a duplicate of, with the number $b$ of coupons per batch equal to the number $n$ of coupons. I use the occasion to make everyone who's reading this aware of the list of generalizations of common questions, where I collected many of the variations of the coupon collector's problem that keep getting asked on the site. – joriki Mar 17 '23 at 14:27

1 Answers1

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NOTE

OP has modified the question after the answer was posted so that now it is a well known coupon collector problem with multiple coupons in each packet. The answer below addresses what I understood to be the original query

It is always good to change abstract problems so that they come within the milieu of familiar types.

So consider, as a parallel for one bag, that a normal die is thrown $6$ times, what is the probability that you get each number exactly once ?

You will immediately say, $Pr = \Large\frac{6!}{6^6}$, won't you ?

So for $n$ colors, for any particular bag, $Pr = \Large\frac{n!}{n^n}$

and applying linearity of expectation, we just multiply the above expression by $b$, the number of bags.

true blue anil
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