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
(In reply to
my result by Cheradenine)
According to Cheradenine's solution, if the previous expected value of the number of unique cards is 198 and I open a pack of 5 unique cards, the expected increase in the number of unique cards is 5 * 2/200 = 0.05
Let's assume the current number of unique cards is known to be 198 (very different than the expected number being 198). For the first card that I remove from the pack, the expected value is 2/200. For the second card, the expected value is 1/199 if the first card was a hit, or 2/199 if it was a miss. For the third card, it's zero if both of the first two cards hit, 1/198 if one did, or 2/198 if neither did. And so on. I don't
think you can just add 2/200 five times to get an expected value for the five cards.
And, we don't know that the current number is 198. We know that there is some probability of 200, some of 199, and so on down to the smallest possible number (5 for problem A) such that the sum of n * p(n) from n=5 to 200 is 198. I don't
think the expected number of new cards given this set of probabilities for the current number is the same as the expected number when the current number is known to be 198.
This is an interesting way to try to solve the problem, and certainly makes it much easier to solve, but I'm dubious that it will give the right answer.
re: my result
