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Suppose you have $k$ black balls and $X\cdot k$ white balls.

The procedure start with you having a bag containing $y\le k$ white balls (e.g. $k+1,\ldots k+y$).

In every iteration:

  1. A single white ball from the bag is replaced by a black ball from outside the bag.
  2. Up to $m$ black balls from the bag are replaced by random (either white or black) balls from outside the bag. Only black balls are replaced, and amount replaced is the minimum between the number of black balls and $m$.

The process is called successful if at any point there was at least one white ball in the bag.

  • What is the probability (as a function of $k,X,y,m,$ and the number of iterations $n$) that the process will succeed? An upper bound will also be highly interesting !
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  • $\begingroup$ Do you really mean that when the bag starts with white balls, the process is a success? And that if there isn't a success, then step 1 could never be used? $\endgroup$
    – Douglas Zare
    Commented Sep 12, 2015 at 16:36
  • $\begingroup$ @DouglasZare - Perhaps I was not clear enough. Only if throughout the $n$ iterations, there was always at least one white ball, then this is a success. i.e., if there were white balls in stages 0-20 and 22-$n$, this is still a failure. $\endgroup$
    – T J
    Commented Sep 12, 2015 at 17:53
  • $\begingroup$ The process can fail for example after iteration #$y$, if all the randomly chosen balls were black. $\endgroup$
    – T J
    Commented Sep 12, 2015 at 17:54

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