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We are collecing stickers in chocolate bars and whenever we open a bar we get a random new sticker. There are many different stickers and we try to collect them all.

We open the first bar and get a sticker. We open the second bar and we get another sticker, but there is now a chance that it's the one we already got. Doubles are thrown away. As we collect more and more different stickers, the chance gets worse and worse.

So if there are a total of $N$ different possible stickers and we already got $n$, how much chocolate bars $d(N,n)$ do we have to open before we get another one, i.e. the $(n+1)^{th}$ sticker? From this it should also be possible to compute the total number of bars we have to open (sum of average openings).

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This is the coupon collecting problem. – David Mitra May 29 '12 at 15:22
@DavidMitra: Ah okay, thanks - make it an answer and I'll give you green. – NikolajK May 29 '12 at 15:27
@NickKidman Now that sounds foggy... – Pedro Tamaroff May 29 '12 at 15:34
up vote 7 down vote accepted

This is one of the basic problems in probability known to many as the Coupon collecting problem. The Wiki link has a complete solution.

Incidentally, you should be asking about the expected number of bars to open to obtain the next, not already collected, sticker, not "how much chocolate bars $d(N,n)$ do we have to open ...".

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