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A box is filled out by $1,000$ balls. The box can be thought of as containing $V$ sites and $V$ balls, with $V=1,000$. The box is repeatedly shaken, so that each ball has enough time to visit all $1,000$ sites. The ball are identical, except for being uniquely numbered from $1$ to $1,000$.

What is the probability that all of the balls labeled from $1$ to $100$ lie in the left hand side of the box?

What is the probability that exactly $P$ of the balls labeled $1$ to $100$ lie in the left hand side of the box?

Using Stirling's approximation, show that this probability is approximately Gaussian. Calculate the mean of $P$. calculate the root mean square fluctuations of $P$ about the mean. Is the Gaussian approximation good?

Any insight is greatly appreciated.

  • 1
    Imagine that the balls are given random positions one at a time. So first ball number 1 is placed in a random position. Then ball number 2 is placed in one the 999 remaining positions, and so on. This point of view makes the problem easier.2012-10-23
  • 0
    I understand that, but don't know hat the probability of finding balls 1-100 on the left hand side would be... 500 balls on lhs, need 1-100. I don't know how to set up the distribution. And then how this distribution would change for P. i understand the question, just can't set up the distributions properly.2012-10-23
  • 0
    What are these "sites" mentioned in the problem?2012-10-23
  • 0
    You can think of the box containing V balls and V sites, with V=10002012-10-23
  • 0
    In your suggested comment, you were taking "on the left" to be the extreme left. The rest of the problem suggests that it should be "on the left side of the box". I have made that more explicit.2012-10-23

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