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Homework Help: Probability: Infinite marbles placed in, and selected from an urn

  1. Sep 12, 2010 #1
    1. The problem statement, all variables and given/known data

    I have a countably infinite set of marbles numbered; 1, 2, 3,..., n.

    I also have an urn that can hold an infinite amount of marbles.

    I then place marbles 1 and 2 into the urn, and remove one of them with the following probabilities:

    The probability of removing a marble is proportional to its number.

    So, the probability that I remove marble 1 is [tex]\stackrel{1}{3}[/tex], and that I remove marble 2 is [tex]\stackrel{2}{3}[/tex].

    Once a marble is removed, I then place marbles 3 and 4 into the urn. I

    Now if marble 2 was removed earlier, then marbles 1,3,4 are in the urn. The probability of removing any of them are now respectively, [tex]\stackrel{1}{8}, \stackrel{3}{8}, and \stackrel{4}{8}[/tex]

    I keep adding and removing marbles as above, in order of their number.

    I am asked to show, that there is a positive probability that marble 1 remains in the urn forever.

    2. Relevant equations



    3. The attempt at a solution

    Not quite sure how to pin this down. Any help is much appreciated!
     
  2. jcsd
  3. Sep 12, 2010 #2

    Dick

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    Homework Helper

    Ok, so the probability marble 1 survives the first pick is 2/3. The probability that it survives the second is 7/8. At this point you have a choice which one to pick which is not 1. Pick the one which has the least probability to be picked which is not 1. That would be a lower bound for the probability that 1 will never be picked, right? I haven't tried to show the resulting infinite product is positive. Can you? That should get you started.
     
  4. Sep 12, 2010 #3
    Thanks for the lead!

    Ok, I'm trying to trace it out now...

    Wouldn't the 2nd round have to be conditioned on the 1st round, so P(surviving 2nd round) would be:

    [tex]\stackrel{2}{3}*\stackrel{7}{8}[/tex], no?

    edit: n.m., I see your point, gonna set it up as:

    [tex]\prod_{n}^{}\mathbb{P}\left\{A_{n}|A_{n-1}\right\}[/tex]
     
    Last edited: Sep 13, 2010
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