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I So I flip 10 coins.. (re: limit of infinite? series)

  1. Mar 14, 2016 #1
    Originally from the statistics forum but am told this is more of a calculus question.

    I flip 10 coins, if any of the coins land on tails, all of the coins split into 10 new coins and I flip them all again. I keep doing this until a round where every single coin lands on heads. Can I expect to ever stop flipping coins as the number of flips goes to infinity? (and followup question: if so, on average, how many flips would it take me?)

    I think we've managed to at least state the problem mathematically but am unsure how to go about deriving an answer.

    From the other thread..

  2. jcsd
  3. Mar 14, 2016 #2


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    If I understand you well, you want the probability of 10 heads in a row ? One in ##2^{10}## ?
  4. Mar 14, 2016 #3
    No that is just the first round, if the first time I flip any of the coins land on tails, then the next round I will flip 100 coins, and so on.
  5. Mar 14, 2016 #4


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    I see. The number of coins multiplies by 10 every time you get a tail ? And all of them have to end up heads ?
    So ##2^{-10}##, then ##
    2^{-100}, \quad
    ##etc ?
  6. Mar 14, 2016 #5

    I flip 10 coins, if any of the 10 are tails, then I go and flip 100 coins. If any of those 100 are tails then I flip 1000 coins. If during any iteration all of the coins I flip in that iteration land on heads then I stop and my task is complete.
  7. Mar 14, 2016 #6
    I suspect that I'm really asking if the series converges? And if it diverges I'd expect to get all heads an infinite number of times if I kept going, and if it converges then I'd expect to get all heads a finite (and possibly less than 1) number of times if I continued on forever?

    Or is making the jump from the value of the series to a statement of probability unfounded/completely wrong?
    Last edited: Mar 14, 2016
  8. Mar 20, 2016 #7


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    You stop after 1 flip with probability 2-10, after two flips with probability (1-2-10)*2-100 and so on. The total probability to stop converges to a value extremely close to 2-10=1/1024, and below 1/1000.
    The probability that you ever get "all heads" is also small (below 1/1000, and dominated by the first flip).
  9. Mar 20, 2016 #8
    thank you!
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