Probability of a complete set.

Hi...
OK, say I have a "deck" of cards, say red, blue and green cards.
The deck is infinite. and I choose three cards from this infinite deck.
I calculate my probability of drawing every possible card (Red, Blue AND
Green) as
3! * (1/3)^3 = (2/9)

where 3! is because the order is not relevant, the 1/3 is the probability
of drawing each possible card, and the ^3 is because there are three
cards that I've drawn.

That seems to work experimentally, so I'm satisfied with that reasoning.
But, I'm having trouble expanding it to 4 (or more) cards drawn.

Empirically, I can enumerate the 81 possible hands that I can draw,
then see how many of those have Red Blue and Green, (36) and
compute
36 * (1/3)^4 = (4/9)

five cards becomes
150 * (1/3)^5 = (50/81)

Both of which work experimentally, so I'm satisifed that the _technique_
works. But, it doesn't help me to generalize to an N size hand. (Basically,
what I'm trying to figure out is (for instance) how many packs of (say)
baseball cards will I have to buy to be (say) 99% sure that I'll have all
the cards in the set. I'm starting with my simple color cards as a starting
point to do that calculation... Is there a way to calculate this without
resorting to enumeration? (Which is fine for 3 types of cards, and say
5 cards, but, might not be feasible for 72 types of cards and 5000 cards))

(My guess is that the formula will be something like
(1/t)^n * f(n,t) where
n is the number of cards drawn,
t is the number of _types_ of cards and
(f(n,t) in my little example is the number of ways
to pick n cards such that all t types of cards are
represented) This I think is what I'm not sure how to
calculate)

Any ideas?
Thanks!
Binesh Bannerjee


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