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Trading cards (Posted on 2002-05-03) Difficulty: 3 of 5
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

  • See The Solution Submitted by levik    
    Rating: 4.1818 (11 votes)

    Comments: ( Back to comment list | You must be logged in to post comments.)
    a method of solving problem B | Comment 8 of 39 |
    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.
      Posted by Steve Hutton on 2002-07-31 13:51:49
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