TITLE: How Many Harry Potter Chocolate Frogs Must I Buy?
QUESTION [5 upvotes]: Each Chocolate Frog comes with one collectable illustrated wizard card (very cool and not dorky at all, honest). There are equal odds of each card being in a pack (i.e., they have all been produced and distributed evenly). How many packs must we buy in order to have an 80% chance of having obtained all 12 cards? How about 90%? Thanks.

REPLY [2 votes]: This website on the coupon collector's problem under item 11a (adapted to 12 items from 6) states that the chance you complete your set on purchase $n$ is $\sum_{j=0}^{11} (-1)^j{11 \choose j}\left( \frac {11-j}{12} \right)^{n-1}$ but gives no derivation.  You can add up starting with $n=12$ until you get to the desired success probability.