A trading card series has 200 different cards in it, which are sold in 5-card packages.
Each package has a random sampling of the cards (assume that any card of the 200 has an equal chance of being in a package).
On the average, how many packages will need to be bought to collect the complete series if...
A: all the cards in a package will always be different
B: a package can have repeats
I don't think there is an easy way to solve this problem. Here is a gross way. Start with a simplified version of the problem:
You draw one card at a time. What is the expected number you must draw to get a complete set of 200.
Let m equal the number of cards so far, n equal the number of unique cards so far, and p(m,n) equal the probability.
p(1,1) = 1
p(2,2) = 199/200
p(2,1) = 1/200
p(3,3) = (199/200) * (198/200)
p(3,2) = (199/200) * (2/200) + (1/200) * (199/200)
p(3,1) = (1/200) * (1/200)
In general, p(m,n) = p(m-1,n) * (n-1)/200 + p(m-1,n-1) * (201-n)/200
Because we stop at 200, the probability that I got to 200 on the last card p(m-1,200) is irrelevant.
p(m,200) = p(m-1,199) * 1/200
The expected value is the sum from m=200 to infinity of m * p(m,200).
Since problem B referred to the number of complete sets of five cards, the expected value is the sum from m=200 to infinity of s * p(m,200) where s = m/5 with any fraction (however small) rounded up.
a method of solving problem B
