🚀 go-pugleaf

John Hanna poses the following problem:

> SOGGIE, breakfast of chumps, is having a special promotion.
> They are packing a special 'prize' in each box of SOGGIES.
> There are three different prizes in the collection, distributed
> uniformly among the boxes, and each box contains one prize.
> You, constant reader, buy one box at a time.

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>     c. What is the probability of getting all three prizes in
>        exactly K boxes?

   .....................................................

Offhand inspection (confirmed by a few back-of-the-envelope calculations)
suggests that the probability that a third variety of prize will be
found for the first time in the Kth box is (2^(K-1) - 2)/3^K.  There is,
after all, a 1/3 chance that the Kth box will contain a sample of the
third prize, if the first K-1 boxes each contain one or the other of the
other two prizes.  But we have to exclude the two cases where the first
K-1 boxes all contain exactly the same prize!  This generalizes neatly,
of course, to the case of N different prizes.


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