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Difficult Brain Teaser: Combinatorics Problem on Selection with Replacement

  1. Dec 16, 2007 #1
    Problem Statement:

    Assume that there are n items (numbered from 1 to n) in an urn.
    We select b items from the urn and record their numbers.
    We return the selected b items into the urn and perform another selection.
    We do in total m such selections.

    At the end of the m selections we check the recorded numbers. For every unit number chosen in the first selection, if it is also chosen in any of the rest m-1 selections we have a loss of 1 unit (the loss is 1 unit irrespectively if the same item has been re-chosen more than 1 times).

    Example: n=10,b=4,m=3

    Selection 1 | Selection 2| Selection 3
    item1 | item1 | item3
    item5 | item7 | item1
    item8 | item2 | item9
    item7 | item3 | item10

    In this example item1 is re-chosen in Selection 2 and Selection 3 so we loose 1 unit from it. Also item7 is re-chosen in Selection 2 so we loose another 1 unit from it. In total we have lost 2 units.

    We want to compute:
    a) the formula that gives the average loss that we will have.
    b) the average number of times that an item is re-chosen.

    Actually I think the formula must be of the form:

    Probability(1 unit re-chosen in >=2 selections)*(-1)+Probability(2 units re-chosen in >=2 selections)*(-2)+...+Probability(b units re-chosen in >=2 selections)*(-b)= ???

    Thank you,

    Last edited: Dec 16, 2007
  2. jcsd
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